My thoughts:
I would bet turn slightly bigger, say $1.50 to charge his draws and weaker top pairs and make his call river shove easier.. then I'd shove river. Overbet shove turn is likely profitable as well although maybe he gets away from some weaker Ax hands when we do. If we plan to shove river, conservatively, I think his turn call range is (73 combos)...
ThTs, ThTc, TsTc, 9h9s, 9h9c, 9s9c, 8d8s, 8d8c, 8s8c, 75s, AsKs, AhQh, AsQs, KdQd, AhJh, AsJs, KdJd, QdJd, AhTh, AsTs, JhTh, JsTs, JcTc, Ah9h, As9s, Jh9h, Js9s, Jc9c, Th9h, Ts9s, Tc9c, As8s, 9s8s, 9c8c, Ah7h, As7s, Th7h, Ts7s, Tc7c, 8d7d, Ah6h, 8d6d, 8c6c, 7d6d, Ah5h, 6d5d, Ah4h, Ah3h, Ah2h, AhKd, AhKs, AhKc, AsKd, AsKc, AhQd, AhQs, AhQc, AsQd, AsQh, AsQc, AhJd, AhJs, AhJc, AsJd, AsJh, AsJc, AhTs, AhTc, AsTh, AsTc (equilab)
and his river call range is (67 combos)...
ThTs, ThTc, TsTc, 9h9s, 9h9c, 9s9c, 8d8s, 8d8c, 8s8c, 75s, AsKs, AhQh, AsQs, KdQd, AhJh, AsJs, KdJd, QdJd, AhTh, AsTs, Ah9h, As9s, Th9h, Ts9s, Tc9c, As8s, 9s8s, 9c8c, Ah7h, As7s, Th7h, Ts7s, Tc7c, 8d7d, Ah6h, 8d6d, 8c6c, 7d6d, Ah5h, 6d5d, Ah4h, Ah3h, Ah2h, AhKd, AhKs, AhKc, AsKd, AsKc, AhQd, AhQs, AhQc, AsQd, AsQh, AsQc, AhJd, AhJs, AhJc, AsJd, AsJh, AsJc, AhTs, AhTc, AsTh, AsTc (equilab)
So he's folding 6 combos (JTs and J9s). Say we bet turn as you did that leaves us $2.76 river shove into $4.85. AcKh vs his river call range = 34.36%.
EV of river shove = +$1.14
We can discount some 2pairs, sets and straights on the turn and probably some of his draws, if he plays them aggressively, which makes this even more profitable. I don't like checking river, I don't think there's any air or enough missed draws that he stabs to make it better than shoving.