I think the turn bet size is fine. As played, it's going to leave us with $2.19 in a pot of about $3.10 before the rake if Villain calls. If we go too much bigger, then more worse hands start folding, etc.

We're never folding the turn because we're getting 4.44:1 pot odds and we only need 3.6:1 to make calling profitable if we check/fold every river that doesn't improve our hand.

I think shoving is bad since we stand to be behind his calling range a lot of the time. Calling is great because we know it's super profitable, plus we get to play the river in position with one bet left in the hand in a big pot where it's not very easy to make mistakes.