Hypothetical way ahead or way behind situation

[microbet]

Depending on how you feel this might be heaven or hell (ok, it's hell), but you are going to play an unending series of SnGs with the same people. The villian in this hand is not always going to play the same way, but is going to represent a typical opponent. That is, if 80% of people play the hand one way, the villian will play it that way 80% of the time, etc.

Now over the course of this infinity of hands, on a subset of occasions you will find your self heads up against villian post-flop you will have QQ and the devil will wink at you. Regardless of how villian played preflop, this wink will tell you that half the time villian has JJ and the other half villian has KK. The flop is 269 rainbow. Villian doesn't get any clues about your hand.

What is your plan? You should certainly have a plan for when turn and river miss both of you. Does buyin (presumed skill of the table) affect your play? How does position affect it? Assume there was a raise and a call preflop but the pot is still small enough for either of you to get away from. How does it affect your plan if you were the raiser or villian was? How sensitive are you to the pot size?

It is not late in the SnG, you both have something near your starting stacks and blinds are low. Given this, does it make a difference if, say, it is level 1 vs level 2.

I know the back story is not necessary, but I wanted you to be thinking of this as a general strategy issue and not just one hand. Also, please feel free to just answer some of the questions or just give your general thoughts on these kind of situations or add any factors I have missed.