Firstly it depends....
You have 3-bet the BB, and been called by the button. So what range does the Button call blind 3-bets? If its very small then you may be beat, i.e. he has a hand which he will not release to any bet.
But lets assume he has a really wide 3-bet calling range in position... around 10% He could have pretty much anything (pairs, aces ect)
What is the board like? and what do you have?
I assume as you are talking about a resteal then you have air. Thats OK, its good to be thinking of a resteal.
So provided he has a 3-bet calling range big enough that he isnt just playing AA KK then 1/2 pot is a good amount.
1/2 pot is probably the smallest bet that will put him off a medium hand. When stealing you want to be betting the minimum to get the job done. If he has a hand then he won't fold then you are losing the minimum amount. You need to be asking .. what is the smallest bet I can make that will make him give up all but the strongest of his range. A 1/2 pot c-bet has to work only 33% of the time to show a profit. Whereas 3/4 pot must work 43% of the time and pot 50% of the time. A 1/3 pot bet must work only 25% of the time, but probably has less fold equity in a 3-bet pot.
An ace on the flop is both a curse and a blessing. BEt it anyway. If he was calling with say TT then the ace may well frighten him and give credit to your c-bet.
If he calls, give him some credit and check the turn. If he bets, you are probably beat, unless you have info suggesting that he often floats flops but gives into turn aggression.