What type of cheeks are on the bit? (sorry if this was already asked and I missed it)
When I first got my horse I was borrowing a loose ring double jointed snaffle and found that the corners of his mouth were pink(er) after riding. Nothing major- if he had dark skin I probably never would have noticed. It was the right size, it was just irritating him for some reason. (As a side note, I've never heard of sizing french links up 1/4" and don't see why that would be the case, but I have heard of sizing loose rings up like that to allow a little extra space so the rings don't pinch) I ended up buying him a KK Ultra 2-type which has the same type of action as a loose ring with built-in bit guards and haven't had a problem with it since. Switching to an eggbutt, D-ring, or other type of cheek would probably have fixed the problem, too.