Interesting way to play the hand, after you check the turn I'm betting my ace. I don't like the idea of checking here to get a bet later since many worse hands then KQ can take this line. But with the way the hand turned out I couldn't see someone quit betting an ace on the turn after reraising preflop so I would raise you after checking behind on the turn. I personally think Mike checks behind any kind of hand at all and will bet any ace or better or rags.
Therefore, I like the bet/fold line here because your going to pay off his bet anyway but lose bets when he checks behind with worse hands that would call you like medium pairs. If he raises you I think your always behind here with the exception of a pure bluff, but that takes a lot of balls in limit. It basically comes down to... will Mike chase you HU with absolutely nothing post flop? If so I take the check/fold route. If not and he has something I like my choice in bet/fold because he's gonna cheap showdown when your ahead and bet when your losing.