Preflop : Hero open sb, Vilain flat bb

Flop : AsTc5h
Hero check, Vilain bet 2/3pot, Hero call

Turn : Kd
check / check

River : 2s

In this hand I check with JsTh, vilain bet 2/3pot, I decide to turn it into bluff and x/raise, He calls with Ad2d .... whatever.
My question is more about how i should play my range here.

Vilain's range seems capped :
- no QJ
- A2 (maybe x-back flop)
- some random 22-K2 ??
- Ax Kx
- air (stab on the flop)

My range :
- some AA-KK, A2
- Ax, Tx, QQ,JJ
- some Kxs (with bdfd on the flop)

When vilain x-back on the turn, I assume he is quite capped, so I like ovbet but I have only few nuts hands and no air .... so it does'nt fit well I guess.
If I check and vilain bet, I would like x-raise for same reasons (he is quite capped) but are my "nuts combos" really in my checking range ? (as he will x-back on the river a lot I guess ....)

Any help on how building range in those kind of spots is welcome :)