Flat call on river with "2nd nuts"??

[that_pope]

In a very tight Casino AZ 8/16 game, worst I have seen it in a long time, only one easy spot, the button in this hand.

I raise UTG+1 with two black nines, folded to button, who flashes a card while calling, I see an ace, and he calls, SB folds, and BB calls. BB is a new player, played less than 5 hands, and haven't played with him before.

Flop comes K94 rainbow. Check, I bet, button thinks and folds, BB calls.

Turn comes a 3, BB checks, I bet, he raises, I three bet, he comments and 4 bets, I call.

At this point, I figure he has a set, but it is more likely he has an underset (no preflop reraise) and the fact that there are 2 undersets and 1 overset.

River comes a 7, no flush on the board, and I am not putting him on 56 obviously. He bets, I raise, he threebets, I puke, cry, call, and OBV get shown KK.

Did I overplay my hand? Could I flat call on river with basically the 2nd nuts? Thanks....