PeterH

I think you've got it.
Horizontal cracks within the glaze (acting as reflectors) would affect the colour (by reducing the distance light travels in the glaze) even if the colour is uniformly distributed in the glaze.
The small amounts of colourants in copper-reds might also be significant.
Snowflake Crackle https://ceramicartsnetwork.org/ceramic-recipes/recipe/Snowflake-Crackle
Color
The first thing I noticed when I did color tests with the usual colorants and opacifiers (copper carbonate, red iron oxide, chrome oxide, stains, Zircopax Plus, etc.) was that, as the color improved, the crazing ceased. This was because all oxides have expansion/contraction rates and adding them changed the CTE of the glaze enough to stop the crackle effect. I ran more tests with very low levels of colorants (under 1%), which kept the crazing yet still produced a nice color.
PS Do you have any similarly fired pots which  didn't show this effect? An ice-water stress test might be interesting.
PPS I wanted to suggest that the relative freedom from cracks near the edge of the bowl was a significant clue, but couldn't word it right.
So here is a picture of the effected -- with a smaller crackle size and hence smaller outside ring.
I've been testing a Snowflake crackle glaze. Here it is over coloured slip
https://www.reddit.com/r/Pottery/comments/8jjxxi/ive_been_testing_a_snowflake_crackle_glaze_here/

I think you've got it.
Horizontal cracks within the glaze (acting as reflectors) would affect the colour (by reducing the distance light travels in the glaze) even if the colour is uniformly distributed in the glaze.
The small amounts of colourants in copper-reds might also be significant.
Snowflake Crackle https://ceramicartsnetwork.org/ceramic-recipes/recipe/Snowflake-Crackle
Color
The first thing I noticed when I did color tests with the usual colorants and opacifiers (copper carbonate, red iron oxide, chrome oxide, stains, Zircopax Plus, etc.) was that, as the color improved, the crazing ceased. This was because all oxides have expansion/contraction rates and adding them changed the CTE of the glaze enough to stop the crackle effect. I ran more tests with very low levels of colorants (under 1%), which kept the crazing yet still produced a nice color.
PS Do you have any similarly fired pots which  didn't show this effect? An ice-water stress test might be interesting.
PPS I wanted to suggest that the relative freedom from cracks near the edge of the bowl was a significant clue, but couldn't word it right.
So here is a picture of the effected -- with a smaller crackle size and hence smaller outside ring.
I've been testing a Snowflake crackle glaze. Here it is over coloured slip
https://www.reddit.com/r/Pottery/comments/8jjxxi/ive_been_testing_a_snowflake_crackle_glaze_here/

Thank you for that article @PeterH, copper red is fun and fickle and I’m no better at understanding it after all these years. I can usually manage to get it nowadays though. I gave a shot to trying to get some better photos, but the mojo wasn’t with me and they’re no better than what I posted. When I have a few minutes to play with the camera I’ll try to oblige. 
 @Min, it’s exactly what I first thought of when I saw it, big snowflake glaze.
On another note, some of the reds I get are very much at the surface. It’s the reduction cooling. They make it through the dishwasher fine, but they’ll scuff right off with a light sanding. 

It looks to me like the angular nature could be a crack pattern showing. I've no idea why different regions separated by cracks should show different colours.

As best as I can tell here's what's happening in this situation...
when the liquid slip first makes contact with the dry plaster surface it IMMEDIATELY dries. Then, as more slip continues to flow over it, it forms a "ridge". 
In theory you have two solutions: as I mentioned previously, you can make the point of contact an area not visible from the top OR you can reduce the degree, to which the plaster dries the slip, at that first point of contact. You would do this by wetting the mold before you start casting. (Simply hold the mold, under running water, for just a second.)
Wetting the mold may reduce the height of the bump but it will also slow the casting process. Therein you have a choice.
Porcelain doll artists first introduced me to the idea of wetting a mold before casting. (To reduce sticking.) Subsequently I found it has other benefits as well.

The clay will shrink as it dries, which can be  problem if the balloon doesn't slowly leak/shrink.
I found that a non-intrusive way to let a little air out of a balloon was:
- put a strip of sellotape on an accessible part of the balloon
- rub to get a good sellotape/rubber seal
- prick through the sellotape & balloon with a needle
- let some air out
- reseal the balloon with another strip of sellotape over the first
Try it on a spare balloon first.

The clay will shrink as it dries, which can be  problem if the balloon doesn't slowly leak/shrink.
I found that a non-intrusive way to let a little air out of a balloon was:
- put a strip of sellotape on an accessible part of the balloon
- rub to get a good sellotape/rubber seal
- prick through the sellotape & balloon with a needle
- let some air out
- reseal the balloon with another strip of sellotape over the first
Try it on a spare balloon first.

I high bisque, “low” glaze some of my very thin and fragile porcelain paperclay sculptures that need props to get to vitrification without slumping or warping, so I bisque to cone 8 and then glaze at cone 6, you only need to glaze at least two cones lower than you vitrify your clay at. In order to get the glaze to stick, it is actually pretty easy. Warm up the ware in your kiln or oven or with a torch (being careful with the torch not to stay in any one area too long or you could crack it) and I add CMC and sometimes gum arabic to my glazes. If I am absolutely having a hard time getting the first coat of glaze to stick on I will spray hairspray on the ware or add a little Karo syrup (anything that is sticky and will burn out early) into the glaze. Once you’ve got one coat on, the rest of the coats will stick to the first coat and you will have no problems. I personally spray, but this works with dipping or especially brushing as well. I also use bone china, but like the others wouldn’t recommend it to start with. 

The clay will shrink as it dries, which can be  problem if the balloon doesn't slowly leak/shrink.
I found that a non-intrusive way to let a little air out of a balloon was:
- put a strip of sellotape on an accessible part of the balloon
- rub to get a good sellotape/rubber seal
- prick through the sellotape & balloon with a needle
- let some air out
- reseal the balloon with another strip of sellotape over the first
Try it on a spare balloon first.

Yea, I would try a firing to only 100c first and figure out what's what. Something like 100c top temp, pot F1 set to 5, t.F1 set to 30 min and t.F3 set to 10 min and see what happens.

I am not sure why a kiln would trip it as they work by comparing the power going in and out and if they don't match then it cuts the power. In the UK most panels have a GFCI (RCD) that all the power has to go through first before any breakers. Seems better to have one as if current starts flowing through you to ground then it will probably save your life.

Hi and welcome!
The way you have that question written, I’m picturing you layering clay slabs with toilet paper and then more clay slabs. If the clay pieces don’t contact and blend with each other, they’ll just delaminate, especially once you fire the paper out. BUT. 
There is a technique where you mix wet paper pulp (no more than 20% by volume, but “to taste”)with reclaim slurry, so that the fibres reinforce the clay, and that is indeed called paper clay. You then dry the slurry into a workable consistency, and handbuild with that. The fibres gives pieces incredible green strength and makes attachments quite solid. Paper clay lends itself quite readily to really thin slabs that can be layered with each other, and pieces adhere to each other with just a little paper clay slip applied between. 
There is an article on the parent website here from 2009 about paper clay that was republished recently, and it recommends using cellulose attic insulation. I would personally NOT use this material because it contains fire retardant, and that’s gonna make extra fumes when you bisque your piece. 
This blog post from a former poster, Chris Campbell, does describe how to make your own paper clay quite nicely. If you want to google others, be sure to include ceramic or pottery in your search terms, or you’ll get a lot of paper mache instructions. 
 

A search for "pizza stone" or something similar will throw up a lot of hits.
As you will want to include archived threads in your search, best to use a  browser search something like
site:community.ceramicartsdaily.org baking stone
... which found the following threads, with a fairly common theme - buy a kiln shelf, if only because you cannot get a suitable clay.
Starting with an eerily similar question to your own.
 
 

 

... and another method for reducing texture on your 3D printed master, spray paint.

See Smooth PLA | No Sanding https://youtu.be/rn4edNztaPU
 

Speculation ... based on plaster apparently left adhering to your 3D printed mould.
Could some of your issues be caused by the "texture" generated by the 3D printing process. Are you are pulling the plaster out of the mould "against the grain" rather than normal to it at the points retaining plaster. (Is the issue texture-scale undercuts rather than PVA/plaster adhesion?)
If that's the case, these seem to be some of the ways other people have used to get a smoother/less-textured surface.
Sanding 3D Prints: How to Sand PLA & More
https://all3dp.com/2/sanding-3d-print-pla/
... includes How to Print Smooth Parts
PLA Smoothing: How to Smooth 3D Prints
https://all3dp.com/2/pla-smoothing-a-beginner-s-guide/
UK 3D printing epoxy ads
https://tinyurl.com/ebvhn2ed
https://tinyurl.com/2jvcuvn3
PS The sacrificial mould technique would probably tend to remove the PLA from the plaster in a direction more normal to the local plaster surface and its texture, which might reduce damage. in some circumstances -- but I don't think it would work well at the edges of the handle.
PPS At one point I found an article advising that auto-restart (after power failure) was turned off to get a smoother 3D print, but I cannot find it again.
...  this seems to mention the issue.
Power Loss Recovery Might Make 3D-Printed Blobs
https://hackaday.com/2022/10/10/power-loss-recovery-might-make-3d-printed-blobs/

Just or the record, there are several 3D printing pages at digitalfire.
3D Printing a Clay Cookie Cutter-Stamper https://digitalfire.com/video/3d+printing+a+clay+cookie+cutter-stamper
3D-Printing https://digitalfire.com/glossary/3d-printing
3D Design https://digitalfire.com/glossary/3d+design
3D Printer https://digitalfire.com/glossary/3d+printer
Project: 2019 Jiggering-Casting Project https://digitalfire.com/project/15

I was definitely NOT suggesting that you used bone china, it warps/slumps too much. Just using it as the classical example of high-bisque low-glost approach. If you do read up on it I would concentrate on how they get the glaze to stick to a non-porous body: pre-heating, additives, ...? 
Will you need the puck to be glazed on all surfaces? If so using stilts is one option, another might be to  glaze the "bottom" in an additional lower-fired glazing.

Casting bodies don't need to be very plastic, and can have comparatively low shrinkage.

https://digitalfire.com/glossary/slip+casting
Slip cast ware can dry-shrink as little as 1.5% (compared to 6%-8% for plastic stoneware bodies).
https://www.potterycrafts.co.uk/Products/pottery-powdered-clays/P3101
Shrinkage (w-d) 2 - 4% & Shrinkage (d-f) 6 - 8%
So maybe 4% for bisque to fully fired?
However you could high-fire  to maturity, grind, and then either high or low-fire to glaze -- so zero shrinkage. Bone-china is traditionally fired high-bisque low-glost (it needs to be supported when fired to maturity).
Glazing a mature body presents problems, as it isn't porous, but is probably doable (AFAIK industrially various additives are used).
PS Solid casting is possible, unless the gradual build-up of clay obstructs the inflow of fresh slip. A hockey puck seems a pretty undemanding shape.

Casting bodies don't need to be very plastic, and can have comparatively low shrinkage.

https://digitalfire.com/glossary/slip+casting
Slip cast ware can dry-shrink as little as 1.5% (compared to 6%-8% for plastic stoneware bodies).
https://www.potterycrafts.co.uk/Products/pottery-powdered-clays/P3101
Shrinkage (w-d) 2 - 4% & Shrinkage (d-f) 6 - 8%
So maybe 4% for bisque to fully fired?
However you could high-fire  to maturity, grind, and then either high or low-fire to glaze -- so zero shrinkage. Bone-china is traditionally fired high-bisque low-glost (it needs to be supported when fired to maturity).
Glazing a mature body presents problems, as it isn't porous, but is probably doable (AFAIK industrially various additives are used).
PS Solid casting is possible, unless the gradual build-up of clay obstructs the inflow of fresh slip. A hockey puck seems a pretty undemanding shape.

Interesting questions!
To try for +/- twenty thousandths (a fiftieth of an inch), perhaps taking off the last little bit after the final fire - for unglazed ware - would be an option?
Then the additional shrinkage wouldn't be a factor.
I'm curious what you're making.
Where two (or more) parts have to fit together, one can get fairly close by measuring the pieces when they are in the same state, where it helps that the pieces are about the same thickness, have similar curves, were worked similarly, there's more than one each to pick from, and there's experience, specific trials, and notes to draw upon.
Fitting together may be a simpler problem that meeting your dimension spec.
As for thickness, too thick can happen quickly. Getting the ware to dry well - evenly - without cracking, distortion, and then firing so the thick bits are bisqued all the way through, again, without defect, may be challenging. I'd suggest adding only the amount necessary to allow for grinding to spec.
Any road, given everything is repeatable, you might be able to get close.

Buy a kiln shelve (mulite-the tan ones)and cut it down to whatever size you need on a diamond wet saw-this is easier than any other option

Article on Turner's white with some examples
It Slices, It Dices! Some Simple Glaze Tests Reveal a Ceramic Glaze That Can Do it All (well, almost)
https://ceramicartsnetwork.org/daily/article/It-Slices-It-Dices-Some-Simple-Glaze-Tests-Reveal-a-Ceramic-Glaze-That-Can-Do-it-All-well-almost
.. I notice it says under Colour Development
The second part of our project was to use Turner's White as a base, but just exclude the Zircopax (an opacifier).

A search for "pizza stone" or something similar will throw up a lot of hits.
As you will want to include archived threads in your search, best to use a  browser search something like
site:community.ceramicartsdaily.org baking stone
... which found the following threads, with a fairly common theme - buy a kiln shelf, if only because you cannot get a suitable clay.
Starting with an eerily similar question to your own.
 
 

 

May I suggest that you make a few more birds, a little practice/experimentation goes a long way.
Then maybe try for a hollow bird
How to make a hollow bird form using thumb pots
... Make a small through hole with a skewer to let any steam generated during firing escape.
You can make some bowl-ish items by draping thin class clay slabs over a balloon.
e.g. clay over a balloon https://www.pinterest.co.uk/jazzdd76/clay-over-a-balloon/
... which also mentions a few projects, such as

More balloon ideas at Pottery Ballon Technique https://www.pinterest.ca/black3200/pottery-ballon-technique/

I've been making 3d printed models for molds for glass casting, concrete casting, etc, for about seven years and now delving into plaster mold making for slipcasting. I've never had success making a plaster cast directly from a plastic model. Also, plastic models printed on low to mid-priced FDM printers are always going to need 'post-processing' to smooth out the surface (especially the tell-tale 3d printer striations) and seal it properly.  I use automotive primer/filler paints or Smooth-On's XTC-3d epoxy to fill and sand, then shellac, wax, etc. It's an iterative and time-consuming process but can't be skipped.  Once I print the model (I use Rhinoceros software to design - and check for undercuts within the software), I post-process significantly to get a 'perfect' surface as mentioned, cast a rubber mold (I use mostly Smooth-On Vytaflex 40 urethane rubber), then cast the plaster duplicates from the rubber mold. If I have a concave surface model that is small, say a teacup where it would be difficult to post-process the printed model and/or you don't want to make your knuckles bleed, I take one more step and make a convex positive instead, where I'm then able to post-process more easily an exposed outside/convex surface.  From that I then cast two rubber molds - the first rubber mold from the plastic model, and then the second and final rubber mold from the first rubber mold. When casting the rubber molds, I always use a release agent recommended by the rubber manufacturer. No release is necessary to cast the plaster from the rubber mold.  The process is much work, but it's worth the effort, if not for the learning when the failures happen (usually design-related)!