Once he checks to me on the flop he could just be giving up, i'm betting pot. This generates better fold equity, plus with massive equity i want to get money into the pot as soon as possible, whether it's directly by betting strong myself or indirectly by forcing him to c/r bigger if that's how he plays in blinds battles.
We're certainly not going anywhere when he raises, the question is should we 3bet him right there or not. Reads on him would help here, especially how he values hands in blind battles. Most tags won't value check/raise a top pair to begin with in 6max games, even in blinds battles and certainly not on that board.
So if he's doing that for value he has two pairs, a set or a straight there most of the time. 65, 99, 66, 55, 87. We are a 2:1 dog against that part of his range if we get it in right there, which is not that bad, as there are few of those hands. Plus he might also just bet bet bet those hands, not willing to let a free card that could destroy his hand peel on the turn.
If he's semibluffing he might have worse flush draws, or some kind of pair+draw combo with something like 7c5c. We have flush draws crushed, and still have massive equity against any kind of pair+draw with our overcards outs, so we don't mind getting all the money in right there. Making a smallish 3bet that looks like a rebluff to allow him to shove over is likely best here.
If he's bluffing he might have anything, like naked overcards or something. We have most of those hands crushed because we are ahead with AT-high right now. I don't expect a random tag to just go nuts and bluff again 2 more streets with total air, so i don't mind taking the pot down with a 3bet right now, which also has the advantage of blowing off better hands like AK-high.
Considering the likelihood of each range, i'd say air >> semibluffs > made hands in a blinds battle. For me, along with the metagame advantages of sending him the message that he'd better not try to mess with our BB, i think this seals the deal in favor of just 3betting him to like $8,5 right now hoping he shoves worse hands over for the same kind of metagame reasons. We're not worried about giving him great odds if he has a flushdraw, quite the contrary. If we're behind, we have outs.
Plus if you just call the flop you get into tricky spots with A-high against an unknown, not knowing what the hell he's doing. It's going to be a guessing game.
Like as played, he bet just enough to leave himself a potsized bet on the river. If he's doing that planning on shoving that river, we have good implieds odd to call. If not, like maybe he's just stabbing at the pot again cheaply or value betting a hand he can find a fold with on a club river. We're either calling down with A-high against immediate odds for our draw, or folding what may still be the best hand.
Same kind of problems on a blank river, if he bets we're likely gone as we don't know if we can take him off anything with a raise, our line wouldn't make much sense. If he just checks same thing, we probably have to just check it back and hope he was bluffing, as we can't be sure he folds whatever he has even if we shove, he might be c/calling hoping we bluff our busted draw into him.
It's just a pain in the ass without some kind of read or history really.