There is one rare one just called flawless battle stone that is supposed to be boe, but that is it. The rest (type specific ones) are all bop. I am not sure if the boe one can come from randomly battling pets or not, it might be bag only.

I've just been beating all 3 and getting stones. But you only get the stone of the first pet.

I've only done around 15 battles so far but I've gotten two stones so far. A Flawless beast one from the jade forest and a Flawless critter one from the valley of the four winds.