KK flops top set, death card on the turn

[chh]

Daily 15k (10 min lvls, 10+1)

Villain is very successful at these lower limits and definitely the toughest player at the table (TAGish).

PokerStars No-Limit Hold'em Tourney, Big Blind is t50 (9 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

Hero (t4760)
BB (t1315)
UTG (t1270)
UTG+1 (t890)
MP1 (t2965)
MP2 (t4060)
MP3 (t1865)
CO (t1395)
Button (t2530)

Preflop: Hero is SB with K, K.
3 folds, MP2 calls t50, MP3 calls t50, 2 folds, Hero raises to t250, 1 fold, MP2 calls t200, MP3 calls t200.

Flop: (t800) K, T, Q (3 players)
Hero bets t500, MP2 calls t500, MP3 folds.

Turn: (t1800) A (2 players)
Hero checks, MP2 bets t1000, Hero calls t1000.

River: (t3800) 6 (2 players)[
Hero checks, MP2 bets t2310 (All-In), Hero ??

Should I bet more on the flop here? I mean obv I'm giving too good odds to some draws, but e.g. the flush might make his hand and still be 2nd best.
On the turn I'm obv just calling looking for the board to pair.
On the river... Can I ever expect a thinking opponent to be pushing with less than a jack? He was taking a little time but that might be because of the complexity of the board.

fire away