[5NL] TT check back river?

Bet. That's the best river card you are going to get (other than a T) for betting 3 streets. Against this villain, it's not even close, as he will most likely still call loads of worse hands.

Or at this point is everything I beat just folding if I bet?

Depends on betsizing. But you can definitely bet an amount here that gets calls from worse.

Great river card for us so def. bet. The K + him checking is less reason he has a king plus we counter any of his 2 pairs that he just c/c the turn a small% of the time plus the fact he's probably been behind the whole hand anyways.