Welcome to the BrickWood Forums! You’re at a great place in your build.
You are correct in understanding that you’d only paint the stucco if you’re not planning to apply a finishing veneer. You can always change your mind later, but then you’d have to strip the paint.
This thread has some experiences in it. In particular, you can see pictures of an oven where the builder tinted the stucco itself rather than painting it. The advantage would be avoiding any danger of paint flaking off and leaving a white “chip” in it.
But 30 days to cure seems really excessive. You’re not applying this as house siding, and very little of it is truly vertical. I think you’d be safe with 4-5 days to allow it to set up and cure. One builder in the thread above used masonry paint, which would be a good choice; or you can use a good grade of exterior house paint.
I don’t recommend doing this after you do your curing fires. You’ll have soot contaminating the surface. Washing off the soot involves some brushing and chemicals, because of where it’s landing. And if that’s not enough, if your stucco was still a little soft, it means that the soot will embed itself in the soft stucco, becoming like a little dinosaur fossil as the stucco hardens.
Let us know how it goes. You’re so close to those first fires!