I sometimes make mistakes like this too. Bet near full pot on the flop and the long term results will be good, I think he checked to you on the flop with the intent of CRing you all in which would have been awesome, you gave the board a chance to scare card up and really screwed your chances of getting any value. Slow playing is one thing but checking behind and letting NO MONEY go in on an entire street is just not good.

This is absolutely the wrong time to check behind on the flop. You KNOW 100% he has an overpair here. Any opponent with less that 2% PFR that min-reraises has AA or KK 100% of the time. If he is passive, you can't count on him betting for you.

So, bet all streets! I would bet $3.5 on the flop, $7-8 on the turn and push the river. That way you double thru this guy, rather than making only 1/3 of your stack. Huge lost value there. He's probably not going to fold...he is slowplaying his KK thinking you will bet into him with worse.