I have been making long neck bottles and vases. I have been unable to get them to stay put when trimming. The bottom of the vase is larger than the top, so they tend to not hold steady even if I use clay to hold it still.
It sounds to me that the vessel may be too dry to stick to the bat or your tools might be dull and you're compensating by pushing harder than you think.
I make 16th century copies of Bellarmine jars and 17th c. jugs which should be similiar in shape to your long neck bottles. As soon as my vessels come off
the bats I palm in the bottom and cover with plastic. I trim from the top down, while the clay is still "tacky". I allow the weight of the jar to hold itself in place. If by chance I need tabs of clay to hold the foot in place, its usually slip from my bucket wiped along the foot and bat. You just need something to hold
the vessel in place temporarily. I don't believe in Giffin Grips for a couple of reasons..
1. They cost money. 2. They only guarantee that the base is centered. And since I trim everything right side up, that doesn't seem to work for me.
And if you turn the vessel over to trim a foot, then only the rim is centered.
I do have to re-center 2 - 4 times as I trim from top to bottom, but that is ok since otherwise, I'd trim one side thinner than the other. I have used Giffin Grips and chucks in the past, but now see no use in them.
Also, check the dullness of your trimming tools.
Keep us posted,