I raise that turn for information and to build the pot. KQ is possible, but I don't buy it the way he is leading. I am building the pot for now and likely to get as much in the middle as possible.
The way you played it: On the river, I raise his bet to $12. I think he'll call another $7 to have a shot at $29, if he has a queen. Otherwise, you don't have to show your hand. You also leave enough behind to fold to an All-In re-raise.
I think you may be playing this one a little scared. KQ isn't that likely, given your read.