So, our villain here opens in middle position, and calls a check raise on the flop. At this point his range was very inelastic. I think his range once he call your raise there is TP any kicker, 2nd pair strong kicker, and all sorts of weakish draws i.e. flush or str draws (combo draws might have played back harder).
I think once you check-raise the flop, he calls, and then the ace falls, his range becomes very elastic. Naked draws are not going to continue without the right price, 2nd pair and any under pairs are probably not going to continue unless they still have a draw (i.e.T8s or KT, QT, etc). What was top pair on the flop is now 2nd pair, and probably shouldn't call a bet, and of course KQ got there.
All of the Tecs say the same thing really in this spot. He will fold to large bets with most of his range, but will continue against smaller bets. i.e. if you bet 1.00 here, he is going to call with his whole range, but if you ship it you are only getting called by the tippity top of it equity wise. What we need to do here is find a bet size that makes it an error for him to call, but he will probably call anyway with the widest part of his range.
That being said, the only thing we are afraid of is a flush draw which is roughly 4:1, so we want to offer him something like 3.5:1, which is right around 8$. This is a small enough bet that he might go ahead and raise us with the top of his range as well.
Of course, this is all imo.