This one is really tough without a read on the opponent. I really don't think he hit a boat, but there's such a massive range of hands that I wouldn't place him anywhere.
I think you needed to extract more information on this hand to get a better read. the PF raise is fine - 3BB is fine with A7 shorthanded.
However, on the flop, turn and river each time you bet progressively less
(compared to the pot) - about 1/2 pot on the flop, 1/3 on the turn, and less than 1/3 on the river. This is a weak lead, which is fine if your goal is to keep the opponent in the pot. But in terms of being able to read him, you're not getting information - any hand that made at least a good pair on the flop is likely to call you down.
My best guess is that he's bluffing
. I wouldn't be surprised to see either two clubs or A-x spades on the showdown. Because you were using a weak lead, your opponent may have taken it for a sign of a weak hand (say, an ace with an unpaired kicker), and is trying to buy the pot from you.
I think you're right to push all-in, and if he actually does have the winning hand, make a note that he must be an extreme slow-play artist.