KK, Value at Risk

[lehighguy]

$600NL 6max

Hero has KK on button

Deep stacks, I have $1000, BB has me covered.
Pretty loose table, but no maniacs.

UTG limps, UTG+1 limps, Hero raises to $36 (6xbb), BB calls, UTG calls, UTG+1 calls.

Flop: 6,5,3 rainbow
Pot: $140

Check, Check, Check, Hero bets $130, BB calls, fold, fold.

Turn: 5
Pot: $400
BB checks, Hero ???

I don't think my opponent is on a draw of any kind. There is no flush, the only preflop hand he could cold call $36 OOP and have a draw is 44.

More likely, I think he has either AA, a set, or a lower overpair (88-QQ).

I can maybe see him committing his stack with QQ. Fish do strange things, but 88-JJ is a little less likely with these deep stacks. If I bet turn again, I'm committing my whole deep stack if he has set or AA. And he may fold thigns I beat.

I elected to check. If he bets river, call. If he checks, bet. I think this controls the pot size and minimizes potential loss.

Net result: He had JJ. He bet 1/2 pot on river and I called. Would a min-raise here make any sense?