This was a rather difficult decision for me as I was even considering folding on the turn. Was leading on the turn a mistake? Sometimes i like min betting river in order to induce a bluff or a "cheap check" but even i don't love my own play here.
I don't like your play here either. I don't hate betting out on the turn, but I'm doing it so I can fold to a raise (absent a read that opponent is very aggressive). I'm indifferent about the river min-bet, but calling the push seems like spewing chips. He's repping AJ+ and there's no reason not to believe him.
Otoh, I hate opponent's play even more. It feels like he must put you on an ace and thinks he can push you off of it, which is an insane thing to think in a $2.50 turbo tournament against an unknown opponent.