[curtains]

What was your original raise? 45?

Anyway I suspect I would check raise. A large % of players will bet a missed AK here, or some other crap, and ok if they have KK and you get called its not the end of the world as you will win about 38% of the time.

Betting out is okay but not for the reasons mentioned above. The plan of betting out and then 3 betting allin seems relatively pointless to me. The stacks are too shallow, and its very unlikely you will be raised by a hand that will fold to an allin. The only purpose for betting out to me is to make AK fold and eliminate the free outs on the turn. Since after betting out, you know you are beaten, why even call the raise? Well you are going to beat AA-KK too often for me to fold here after putting in about 1/3rd of your stack (You will win about 36-38% of the time). Also there is of course some chance your opponent is on some kind of move, or has something like AJ.

Anyway I don't even think it makes that big a difference as to what you do here. You just have to judge how often your opp will bet with AK or AQ if you check. If the answer is that they will bet a large % of the time, then it makes sense to check raise.