KK, huge pot, what now?

KK UTG. I raise, button calls, terrible sb calls, bb 3 bets (he's 44/22/.6 after only 9 hands so pretty much meaningless), I cap.

Flop is T6Q rainbow. checked to me, I bet, button calls, SB raises (and is all-in, BB cold calls, I 3 bet, button folds, BB calls.

Turn is a J, putting 4 different suits on the board. Hero???

I want to bet here because betting is lots of fun, but given the way this guy has played his hand I think his range is very narrow and I can't think of anything he's playing this way that I beat.

What should I be thinking here? I mean, I don't really know this guy, I only have 9 hands with him, so by default I want to call down. Check/calling here seems abnormal, and if I'm behind I have a couple outs (I think he has an ace like every time here so I'll say 2.5 because I'm drawing to a chop a lot of the time), and maybe I'm ahead so check/call at the very least, does leading here accomplish anything at all?

I'll say bet/fold is the line to take, bc maybe I'm ahead, the pot's big and if he raises there is just no way I'm good ever. Thoughts?