Oooh! That's a good one. I like that you didn't reveal how the hand turned out. It's easy the say you would have, should have done after we know the outcome. Of course the right play, might not yield the desired result.
I've said it before, I'm not a big fan of getting them all-in pre because there is always a risk even if you have the best hand. But given the situation you described, I can't say I wouldn't have shoved there. Of course I would have to be at the table and and get a feel for villian's play firsthand. I'm sure there were things (intangibles) that factored into you decision making. With a player showing a tendency to loose call my raises, I might suspect him of making a move to steal the blinds. The 4-bet could be indicative of him actually having a strong hand, but it's likely he just thinks that you're defending your blind and that a strong 4-bet will make you go away. I don't like to shove pre, but I don't like to play scared either. Sometimes you just have to put your man to the test. A-K is pretty good hand to have if you're going to do it. It's hard to say what I would have done. I think I may have shoved there too, based on his prior loose play.