100NL, huge draw just missed, now what?

[eso]

This is from betfair so I got no hand history. 6 max.
Villain has $300, which he's got by busting me twice with, I am stuck $200. I play quite lag, and prolly have a tilty image.
I can only remember one of the times he stacked me: I had AA, raised pre flop. Flop was 939, I fired 2 barrels, then checked the river. No draws had hit and he overbet the pot, $80 into $40. I paid off his 33.
I'm sure the other time he hit big and I couldn't get away from my hand that time too.
But judging from the rest of his play he is not a tag or a nit by any means, and I had labeled him a fish in my notes.

Now this hand, I have ~$160, he covers.
1 limper utg, I raise 97hh to $4.5 in cutoff. Villain calls in sb, everyone else folds.
I may have got the exact amounts wrong because I remember the pot being $12 on the flop.

Flop is 6h Kh 9c. Pot is $12. I bet $12 he calls.

Turn is 8c. He leads out $12, I raise to $36, he calls.

River is 3s. Theres roughly $100 in pot and I have $100 left. Perfect for if I hit. Villain checks to me, and I....?

I put him on any piece of the flop, any draw, he has a wide range. I feel if he had a big hand he would have pushed the turn, as he had perfect opportunity to 3-bet all in.
The only thing stopping me from pushing is my image, he had no idea I called his boat with AA earlier, it could have been ace high for all he knows. So what should I do?

[Shizzle12345]

check behind i think, not much value here.. If you had A9 on the river id bet for value but not with this weak kicker. I sometimes call too here on the turn, if its likely i can get a large bet from him on the river if i hit.

[Daddy Warbucks]

You're not moving him most likely off a K, and your hand has at least some showdown value with all the missed draws out. I check behind.

[scallop]

Check behind. If he views you as tilty then he will call with weaker kings than normal. Plus your hand has some showdown value.

[eso]

Yea he did seem like the type that would call with any king. By pushing I am just taking him off a draw which I am ahead of.
I checked behind and he showed TT. Guess I'm being a bit results oriented, although he'd probably have called with that too.

[JFsports]

easy check behind, if he's missed you win, if he has a K he's not folding. I'd prefer raising a little more on the turn to make it less than a PSB on the river if you hit