[SharkSandwich]

Ben I think his play is pretty horrible. And I'd doubt his 2+2 credentials, unless he had a mad read on you (which seems unlikely).

FWIW I don't really like your play post-Turn. Villain has put in over 200. Over half of his stack. It's very unlikely he's got a hand he's folding at this point. He'll usually be calling off the rest of his money with an overpair (or slowplaying JJ).
Because as has been said, he's done with AK after the flop bet. And yes 120 on the flop is at least 20 too much.
And as I suggested I think you fire one bullet and then don't put any more money in. You've folded what's folding, and are beat by everything that's left.

Though I'm not fond of it I guess he calls PF because he's got position and the aforemention SC (which he figures to be mostly clean outs). That part of his play is probably slighty -EV.
It's his postflop play I find real bad against a non-maniac. Calling the flop isn't good, and I don't like it at all unless he can read you for AK, and knows you won't put in anymore (so he won't have to).
But calling on a pushed Turn here, which is exactly how you'd play the AA/KK you're representing it just SO bad.

Of course it does depend on how often you'd do this with air as opposed to the overpair. And I guess he really thought there was much more of a chance you'd do that than there is...

I guess this all comes from you 3-betting each other rampantyly, so he expects the 4-bet to be light, which it is. And he then expects you to push to try and buy the pot and thinks he's got a good chance you've got air here. That's the only justification I can think of his play though, and it's a reach.