[moot]

I think you absolutely have to raise his flop bet.

First of all, you're the one doing the capping. Personally, I have more hands that I would 3-bet with than cap. For instance, in this situation I would definitely 3-bet your raise with something like AK, but would be less inclined to cap with it.

He could have you dominated, or he could be the one fearing your cap. This is why you have to raise him on the flop. Maybe he didn't want to see you cap it.

Raise him on the flop. If he 3-bets, you can consider dumping the hand. If he smooth calls your raise then checks to you on the turn, punish him for the rest of the hand as he probably has something like AK (unless you get check-raised, then you're probably in a world of hurt).

Smooth calling all the way only to dump the hand at the end is way too weak tight (though I don't totally hate the dump at the end because when the A comes his range of hands that you can beat is pretty slim) (though it still might be worth paying off one more bet) (because by not raising on the flop you have little clue as to what he has) (he could easily have JJ and figure it to be good because you never raised him) (though in that case he might have slowed down on the river when the A came) (I think I'm using too many parentheses).

Raise him on the flop. Go from there. Risking an extra small bet or two on the flop can often save you many down the road.