[sweetjazz]

Couple of points:
(1) AK has a pretty big equity edge if we assume he is 3betting a lot of his broadway holdings. It wouldn't shock me if the guy is 3betting Ax with x going down pretty far (x = 9, x = 7s?). The wider you think his 3betting range is, the more you have to just get it in with AK.

(2) I'm completely mystified about the aversion to taking 60/40 flips. I assume that our hero is properly bankrolled and can rebuy if he loses. If this is live and he can buy-in up to 1/2 the big stack, he can immediately buy-in for $500 if he loses, so we don't sacrifice any future edge. If his next buy-in will be restricted to $200, then that gives some reason to pass on pushing very marginal edges. But you don't want to take that idea too far.

(3) It's irrelevant that the villain will "bleed off his chips relatively soon." If he does that before we get a good opportunity to put our money in, that's too bad for us. Yes, it does mean we will have to take some high variance gambles because we can't sit back and wait for AA, but otherwise it doesn't affect our decision at all.

(4) This is basically a math problem. For any given hand range you assign him, calculate your equity if you push preflop (assuming that he calls his whole range, which seems reasonble based on OP's description) versus taking a flop and folding if you don't flop an A, a K, or a monotone flop giving you a high flush draw plus overs. Again, you can assume that he's willing to get it all-in everytime you don't fold the flop.

I don't know the answer in advance, but you'll find that taking the flop fares better the higher the percentage of paired hands are in his 3betting range. The more Ax, Kx, suited connectors, etc. that you add to his range, the better pushing preflop becomes. It would be interesting to get a rough idea of where the cutoff is where pushing becomes better than calling. I feel pretty confident that calling is significantly better if his range is {22-AA, AQ}, but that getting all-in preflop is much better if his range is {22-AA, A8+, KT+, QT+}. (FWIW, the first range isn't completely ridiculous. Some bad players think that 44 is a better hand than AJ because it would win if no community cards were dealt.)