Youll probably have to go ahead and shove it on the turn. With 44 you might have been able to pass.

But, lets see it from his view. Betting AKcc like that on the turn seems very awkward to me. And, if he's a really good player and has AA/JJ here, wouldn't he go for a checkraise allin here? It's by far the best move. The pot is so big that a bet from you, his opponent, in the range of the pot, would make him comitted. And, if he checks and you do have 44/88 (or JJ if he himself has got AA) you have an obligatory bet pretty much, and if it is in the range of 80-100% of the pot you would have to call his checkraise allin as well, pretty much. The thing is, he cannot know if you have the Axcc by playing the way he is right now. It would be natural for Axcc to just call the turnbet, since hes got nutouts, and might be ahead vs KK/QQ. Then, if a club hits on the river, what is he gonna do? He'd probably have to check/fold vs an allin, since Axcc would perfectly well fit with how the hand was played. The good thing about checking with the intent of checkraising allin here is of course that he would bust all sets since theyd have to bet and would be comitted to call his raise. The only drawback is of course that would totally be in line for Axcc to check behind on the turn, and if a club falls on the river he might have to fold to allinbluff a few times. But, on the bright side of that, if a club doesnt call he can getcalled for quite some money on the river if he bets out and his opponent with the Axcc thinks he might be betting KK/QQ to make his opponent fold an ace.

But, enough ranting of HE should have played the hand, I dont think you can get away of this one. As I said, with bottomset its actually perhaps possible, but in this case, I guess you just have to shove it in. And, if he does have AA/JJ, I don't like his bet for half the pot on turn at all, and neither if he has AKcc.

One last thing to note is that if he indeed has AA, whether he has the A of clubs is of great consequence. If he does, he narrows your hands by quite a lot, and he has to play accordingly. (For example, the above scenario with check/check on the turn would make it pretty much impossible to get paid on the river since his opponent couldnt in any way have an ace.)

(And no, since your checkraise pretty much has no fold equity at all, you obviously dont have to do this move with draws as well. Remember, the checkraise is actullaly a "valuecheckraise", not like a checkraise allin with much larger stacks compared to the pot on the turn, which has to be done with both draws and sets to not be to easily read.)