I'd bet turn. On this river i would x back because on this kind of boards A high will call a lot, maybe even K high, because we either have a pair or something with no showdown value.
And i would x back on T A high and K high FDs

Turn I like a bet too. Most players don't balance their x range in these spots very well, they bet all value and x a weak range. You can get away with bluffing very rigorously and valuebet way thinner. River I agree with wantmoney, board runs out pretty terrible and you have club blockers that you want villain to have (although, given the way the hand played out that is not that important).

In theory, you should just choose the largest bet size possible in these spots (allin) on the river, because the villain's range is relatively capped (I don't think he has an overpair here ever). In this spot, my value range will 4s+, and overbetting the river in these spots have proven quite effective even in microstakes game.

I think it's a fairly large mistake to check twice after the flop call. People tend to overfold when they check and you have virtually 0 showdown value.

bet turn go from there

Do we have any c-bet stats or vs float-bet stats on opponent? I would probably bet either the turn or the river though with your hand given the limited showdown value in a 3 bet pot. I also wouldn't be terribly afraid of getting check raised on that turn.