Villian is unknown

I think everything is standard until the turn, I could 3bet flop but he could be c/r a worse pair with a heart and maybe not call a 3b/shove.

I feel like when he checks to me on turn he has some kind of pair or a hand with a naked A/Q of hearts. Are we shoving or checking in this spot? I think if we check he could turn his hand into a bluff on some rivers or call a river bet off with a weaker pair which he won't do if we bet this turn. If we shove I think he folds most of his c/r range once he checks turn.

Full Tilt, $0.10/$0.25 No Limit Hold'em Cash, Rush, 6 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

UTG: $12.10 (48.4 bb)
MP: $35.16 (140.6 bb)
CO: $37.59 (150.4 bb)
Hero (BTN): $25.30 (101.2 bb)
SB: $42.55 (170.2 bb)
BB: $25 (100 bb)

Preflop: Hero is BTN with K K
2 folds, CO raises to $0.85, Hero raises to $2.75, 2 folds, CO calls $1.90

Flop: ($5.85) 5 J 6 (2 players)
CO checks, Hero bets $3.25, CO raises to $6.50, Hero calls $3.25

Turn: ($18.85) 7 (2 players)
CO checks, Hero has $16.05 behind