^ good point about making the spot hot for him, and preventing a furture hot spot for us. At this point the QQ is very likely ahead, but villain undeniably has outs. So while on Tuesdays, Thursdays and alternating Sundays I might milk the villain, the rest of the time I might shove for that very reason.
However, to bolster the milking scenario, villain will very likely give us some information if we flat the flop, and come along cheaper. This is giving villain some info on purpose. Perhaps timing can induce the perception that we are trapping him. And we do have position on him.
2.5 orbits (20 hands) isn't long enough to get much of a read on anyone, but it is all we have. That and a pretty nice hand. Villain has seen 4 flops, and c-bet 100% of them. This will be his 5th flop seen in 21 hands. Normal flow of cards might (but not so much at only 20-21 hands) suggest he only see about 2 flops, so he is been proactive..
So it really boils down to how much you want to be gambling vs how much you want to play the game of poker. Are you? Were you? Are you always in the same frame of mind at the table? Rationally, either way will work, shove or milk him like a cow. Results over time will favor you shoving in this situation. The milk him concept is my way of keeping bad beats stresses from overwhelming me.
Mathematically, you should probably shove and just win the hand, Problematically, that is probably going to net you less over time.