I think that there are three categories of hands in the villains range
1.) Hands with some showdown value.
2.) Hands that he is simply giving up with.
3.) Big hands (essentially A
x)
I think that the two first categories are pretty big compared to the last one. Betting versus the second category makes little sense. Sure, there are possible hands in the this category that has plenty of outs, especially for a split, but the first category can be pretty huge. Plenty of his "some SD value hands" (like AT - a straight) beat our own SD value.
This calls for putting the ranges in Equilab, and making an EV calculation from there. My hunch is that checking the turn can't be horrible. It may not be the best, but it can't be truly horrible.
If we check the turn, then we don't fold on the river. The option "bet turn" and then go to SD whatsoever seems horrible though (why lose possibly 3 bets). The options bet turn, then either call turn to a raise and fold river to a bet, or bet turn and fold directly to a raise on turn seem to be either or of both of exploitation of the villain or being exploitable ourselves. I don't hate those alternatives though.
Without actually having been to heaven (Equilab), I'd say "check behind on turn, then call a river bet, or make one yourself" is a decent play.
/Johan =