I'm typically a NL player but have been enjoying the symmetry of limit poker lately. Should KK have been played differently here?

PokerRoom 3/6 Hold'em (8 handed)

Preflop: is with , .
BeaverPoker raises, 1 fold, MP1 3-bets, 5 folds, BeaverPoker caps, MP1 calls.

Flop: (7.50 SB) , , (2 players)
BeaverPoker bets, MP1 calls.

Turn: (4.75 BB) (2 players)
BeaverPoker bets, MP1 raises, BeaverPoker 3-bets, MP1 caps.

River: (11.75 BB) (2 players)
BeaverPoker bets, MP1 raises, BeaverPoker calls.

Final Pot: 14.75 BB

Results in white below:
BeaverPoker shows Kh Kc (two pair, kings and sixes).
MP1 has As Ac (two pair, aces and sixes).
Outcome: MP1 wins 14.75 BB.

When he raises you on the turn you should start to get a real sneaking suspicion he has AA after three betting preflop and then just calling your flop bet. He knows were you are at with your preflop cap so he just waits for you on the turn. That said I would have just called his turn bet and check called the river because if he didn't have AA the the next likely holding is QQ. When (MOST) people at this limit three bet preflop you can generally put them on one of 4 hands and by the way the hand went down you can almost be certain that you are going to be looking at AA or a lesser chance of QQ. A fold is probably warranted but those kind of folds are difficult to make and I can't say that I make them very often unless I know my opponent extremely well. I just suck it up and check call the river.