First off, I like the PF 3bet. KJs is plenty strong enough vs. what is most likely a short handed blind steal. The bet sizing seems about right. In most cases this will result in an almost automatic CB Fold. I'm assuming the Villain stats are for the BB and his call was certainly not a part of the plan. No idea what the sample size is but we go with the information we have and his stats suggest a tight aggressive player. Since since a 3 bet call is neither tight nor aggressive you have to give him credit for a real non QQ-AA hand. 88-JJ, KQ,(i'd fold this) AJs+, AK, AQ (I'd fold AQo too but this is .05/.10).
As I said, the BB's flat definitely throws a wrench into your plans and if I didn't hit the flop in some meaningful way you should be done with this hand. I'm not really sure what the best way to approach this flop is but given that you're OOP to 2 players I would probably just check and see what happens. If the check is followed by a bet and a raise then again i would just fold. In fact, if the BB bets at all I just can't imagine that KJ is ever good. 3 bet pots are usually pretty crazy HU, add a player and all of the sudden everyone becomes George Washington, "I cannot tell a lie". Yes, you flopped top pair but this flop is just about as ugly as it gets. If we stick with his likely pf flatting range (and we always should), 88, TT, JJ, AJ all have you beat and will never fold. KQ has an open ender, a flush draw or both not to mention 2 overcards. AQs and AK are about the only calling hands I can think of that you're way ahead of. Only 99 is a total whiff. You could also be facing a slow played nut or near nut hand like AdJd-AdKd or KdQd. At this point you're a slight underdog. Proceed with caution.
You bet out, he just calls, and as expected, the button folds. Given the dangerous texture of the flop, I think we can eliminate 88, TT, JJ and probably AJ from his range as he would probably raise to protect his made hand, which leaves him only with the drawing hands, KQ, AQ, and AK. The flop is Ad. What an interesting card. Given his likely range, there were exacly 40 combinations of hands he could have had, the Ad eliminated 7 and his equity shot up to 89%. BUT, there are precious few that can proceed with the hand undaunted. In short, only a hand containing a diamond can continue. 3 combos of AxKd and 4 combos KdQx. If you throw in the non nut Qd hands of which there are 7 you are left with 14 hands that can stand the heat and 17 that will probably fold. If you're going to bluff this is the time to do it, on the turn, not the river. If you check the turn you can never credibly bluff the river vs an almost certainly made hand. It's now or never and I say go for it. This is an easy bet fold and a 1/2 to 3/4 pot bet seems about right. Anything less and you might get looked up for shits and giggles, anything more is a waste of money as anything that will call you on the turn will call you on the river.
As played I don't think you can bet the river as it will look way too bluffy. Villain is well aware that his check back on the turn looked weak. His best play is to call any bet and raise the nuts.
Sorry for the long windedness (is that a word?). I spent the morning watching the news. Thinking about this thread was a nice way to get my mind off the faces of those little angels.