Live hand vs fish A2o, $1/2 NL $40-$100 max

I was playing a live hand a few days ago at Seneca Niagara, $1/2.

I bought in for $100 (after getting outdrawn) and I was dealt A2o on the button.

A player that everyone tagged at the table as a fish, limps in from UTG2.

Three other players call. I'm sitting on the button with A2o, usually not a great hand, but against a fish who will get it all-in with 88 with overcards on the board, not a bad hand.

So I raise it up to $14, hoping to isolate the fish.

I believe it came three way, fish called, another player called.

Flop comes AK9. Pot is roughly $52 after rake now. Fish leads out for $25. Other player folds.

I decide to call.

Against a competent player, I'm folding here. But knowing the fish could be playing TJ, TT, JJ, QQ, any king, any ace like this, and the fact the board has two high cards on it, his ace kicker is likely to be counterfeited. I decide to call, and see what he does on the turn. If he shoves the turn, I'm probably folding, unless of course I hit a 2.

Turn is an A.

Now he leads out for $50.

Tough spot for me, because, that means he probably doesn't have a weaker hand, he probably has an ace here. There's $102 + 50 now in the pot, and it's $50 more for me to call, but also basically putting me all-in.

He didn't raise before the flop, so I'm inclined to think he doesn't have AQ, AK. Maybe AJ, but also unlikely. I saw him raise pocket eights earlier, so if he had a strong hand like AQ/AK, I'd think he'd raise it, but of course, the mindset of a fish is pretty unpredictable.

That said, I'm chopping with any other A2 and A3-A8. I have 3 king and 3 nine outs to chop against AQ, an additional 4 queens for outs against AJ (7 outs) and of course the three 2s to take the entire pot. And 11 outs for a split against AT.

But really, what I was thinking was, it was $50 more for a $150 pot, and I had to win 1/4 of the time to break even. So I had to call.

He ended up showing AT, and I think a 7 hit the river, so his kicker played.

I'm trying to determine whether my thought process anywhere along the line here was wrong.

Thanks