As there are no obvious draws on the board, BTN probably has something here. Question is, what is it and would he lay it down to one more bet?
I think he won't lay it down to one more bet. At least, most players who will call a turn bet with an underpair here are also going to call a river bet. Not unless he has some pair less than 77, leaving 55 and 33. as the only hands he may have (that beat me) that he would now fold.
Worst case is he flopped a set of QQQ, 888 or 333, and he's been slow playing me. Most slow players raise the turn, but not all, so its a concern. I've also seen players do this with AA and KK, but not too often.
If he had a weak Q, like QJ, QT or Q9, he may just be calling down. A good player raises the flop there, but I try not to play good players, so he could be calling all the way. If I check here, he likely bets. He also calls my bet.
Almost all players raise the flop with AQ or KQ there.
A8 calls a bet. A2 might lay one down as all it beats is a flat out bluff from me. He has to have me on exactly AK to call with A2.
JJ, TT, 99 also all possible.
He could also be calling with AK or possibly AJ, and may lay them down to another bet.
So, of all the hands he might have, I beat very few, and of all the hands he might have, he lays down very few if I bet here. There's 6.25 big bets in the pot. If I bet another into it, I need to win about one time in seven to make it worth my while.
I don't think I have much fold equity, but I think it more likely he will fold than raise, although it is MOST likely he will call. If I check and he bets, I am beat, but I
really want to see his cards.
This is not an easy decision! The good news on that is that you're not making a horrible mistake either way.
Here's what I do: If it's early in the session I check-call. I do this so I can see his cards and to give myself a weak table image. Someone will notice I don't like to bet the river with an unmade hand. This gives me a chance to fire that third bullet later in the session and have a better chance to get the fold I want because they may not put me on AK again.
If it's late in the session and I have check-folded here before, I fire away!