I am playing 1/2 NL at a full table. I' have pretty good position with the two people that know what they are doing to my right.

The villain in this hand sat at the table about 90 minutes ago and is playing very well and LAggy. He handles chips well and has taken down a few pots by either A) Iso/raise limpers IP and cbet or B) Limp/call OOP or call IP and used a raise OTF to take down a pot. He has reraised flops much more often than you would expect him to actually have a hand and has been caught raising IP with trash. Definitely a cut above most 1/2 players.

The Hero has been doing much of the same but with a tighter range and isn't involved in nearly as many hands.

For the hand...
Hero is in the cutoff with JcJd, raises to 12 (fairly standard size) over 3 limpers including the Villain. The BB calls and so does Villain after a short pause. Villain has around $250 and hero covers.

Flop($40): 568r. Checks around, hero cbets 27. Obviously this does not really hit my PF raising range, but Villain knows that and so can call with almost anything. BB folds, Villain c/r to $80. Hero does ???

Obviously this can be a bluff, especially considering the villain history, but the board is rather scary. I really don't think he has a set, as he wouldn't have hesitated to call after I raised PF with a pocket pair.

If I call this, there is ~$200 in the pot and there is less than a PSB remaining, so this is basically an all in. Can I call the flop and fold the turn if shoved on? What would you do here?

I prefer a slightly larger bet preflop. As played, fold. You bet the flop hard telling the villain that you liked your hand and he re-raised you saying that he likes his even more. You are repping an overpair and he is not scared. Sure he could possibly have 87 or 67 but in general if he doesn't have you beat then he's not as good as you think. BTW, this is a shove/fold spot. Calling to fold the turn would be really bad.

He is going to barrel his entire range OTT so you absolutely cannot call and fold to a turn bet. Based on description, he probably has pure bluffs in his range in addition to semibluffs. He could also have 2pr+ and crush you. If it was a two-tone board, I may shove but here, I think a shove just gets snapped off by 2pr+ and folds out all bluffs/semibluffs (except maybe 77). I wouldn't fault a fold but could also call with the intention of getting it in on any turn.

I like shove here.

Hands that you're ahead of that can call:
77 (6 combos), 57 (12), 67 (12), 87 (12), T7s (4), A7 (16), K7s (4, but discount to 2), 99 (6), TT (6) = 76 combos. If the draws fold, that's fine too. Bluffs like two overs will fold (which argues for a call), but I think draws are more likely so you should shove

Hands that beat you. 55 (3), 66 (3), 88 (3), 56 (9), 58 (9, but discount to 5), 68 (9), 97 (16) = 48 combos. Ruling out QQ+ due to preflop action.

That said, likely in the hand I would just call, planning to call anything, except maybe a 4 or a 9 (then try to soul read). (That's why I should shove.) All of the above relies heavily on the read up to this point, assuming he is playing draws this way. If not, then fold.

Edit: The above doesn't even mention the one pair hands in villains rainge. A8, A6(s?) (ok, that would be a total spaz), T8, but at least A8 seems in this Villain's flop raise range. For that part of Villains range, it would be better to call, but in the shove versus fold analysis, they add to the value of shoving.