His calling range isn't the only consideration. We make lots from the dead money when he folds too. I'd say IF either ships light over a 4bet then fine, go ahead, but typically a flat/small 4bet vs a squeeze is the nuts and most regs know this and will fold stuff like 99. IME they never fold it when you shove though since people do that so often with small pairs and random jams when they think people are ****ing around. 4nl players don't squeeze J9s because they think it's a good spot. They 3bet TT because they have a big-ish pair and heard aggression is good, and they can't fold to a shove.
I shove, as played.
That said, I just 3bet pf. %100 f3b over 300 hands from a tighter player is like hands? Maybe? Not enough to assume 3betting QQ to stack it isn't going to be massively profitable. I also tend to hate reading about flats in these spots when I know hero should be 3betting villain wide to manipulate villain and get more info on his 3bet/4bet game, and thus 3betting QQ should be a slam dunk since we should have history.
If you're not sure about whether 3betting QQ is profitable co/btn vs anyone looser than like an 8/7, then you should probably just be 3betting it, willing to stack.