If he didn't eat them, what would he do with the ones that he won? Would he lose on purpose just to be rid of them before they grew into cats and weren't worth anything? Seems like a waste of a perfectly good food supply especially after going through all the effort of winning them.

Fenchurche below reminded me that he seemed to be stalking rats in S7, either just because he was crazy or that's how he was feeding himself while in the school's basement. I had completely forgotten about that.