I would have preferred to fold pre, if not that then 4!. You have one of the worst aces you can have and you're HU so there's not as much IO for making a flush.
As played, if you had a hand bigger than TP, you would have raised the flop. So he should never believe you when you ship the turn. I suppose he could have a junk hand like 44 and fold, but it's not likely. Our flop calling range is KQ, maybe a couple of AK, some hands like ATs/JTs, and draws. KQ wouldn't raise here, AK wouldn't raise here, AT wouldn't raise here...see where I'm going with this?
As to what he can have, it's TP or better most of the time, which won't fold, or a hand that completely missed like 87s and we don't need to get him to fold. Only the few times he has a baby PP is a turn raise profitable.
I think you either play this fast on the flop or not. Trying to go slow then fast on a wet board doesn't add up.