It would be stronger if you made adobe instead of just raw clay. I don't actually know if it would be workable enough, though.
Make maybe something along the lines of 70% clay, 15% sand, 15% (quite finely minced) fiber? This is not what they use for adobe, but that's because they make big crude rectangular bricks and walls, not little pinch pots, so I'm guesstimating it closer to the raw clay.
You could then use very small amount of heat to burn off surface straw that would mess up your finish, then paint. And it should be quite a bit stronger than just greenware. IF you can actually still make what you want, which you might not be able to. If it is workable enough, though, I'd try to increase the proportions of sand and/or fiber until it just barely isn't, and work near there.
Also, firing clay in the coals of a barbecue is one of the few things I actually have real experience with and have done some decent amount of reading on (well, pits more so): low fire earthenware clays will generally hold up BETTER than fire clays or porcelain, due to usually higher thermal shock resistance. So your backyard clay might work in the barbecue stressfully quick firing while fancy white store bought clay might not, somewhat counterintuitively. Tempering the clay will also help. Even the same recipe above would help quite a bit to prevent damage from rapid, short firing: the sand is just grog, and the fiber will burn off and allow porosity and quicker heat penetration and thus more even heating and less breakage.
When I have done this, it's still brittle (as I have been led to understand in another thread, probably mostly because I didn't hold it at heat long enough and/or heat slowly enough), but it's WAY better than greenware still. Probably takes 3-4 times as much force to break the same piece "fired" this way than its greenware equivalent (without tempering)