First of all I'd squeeze preflop. Even if he's not too lose we're happy to take down the pot, AQo doesn't play particularly well 3way OOP.
As played I like to lead the flop also, but as a matter of fact he puts us in a pretty tough spot once he raises the bet (the lead is still good because he calls most of the time if he decides to continue).
A 3bet is not too good for the reasons you already stated (mostly we're getting it in when we're crushed, see also my list of valuecombinations below), but calling also looks pretty strong.
After thinking about it for a while I'm getting more and more confused about what the best option is for the turn:
If we check the turn he is likely to check back QT, he might still bet AT. He is probably also checking back with most bluffs since our call OOP on the flop looks so strong.
That should incline us to bet.
But if we go through his valuecombinations I'm not sure if we can really bet out profitably (even presuming we b/f):
Assuming he raises every 2pair+ there are 6 combinations of QT, 6 of AT, 3 of TT and 1 of AA and QQ each.
Also there are 16 combinations of KJ.
So overall we beat 12 combinations and we are beaten by 21.
But he may be more inclined to just call QT on the flop, so imo it's safe to assume that on average he has about 8-9 combinations we beat.
So combining the information above I got to the conclusion that we actually turn our hand into a bluffcatcher on the flop and should be consistent with that line.
Therefore on the turn either we think he bluffs often enough (and bets worse hands for value) to make a c/c profitable, or we just c/f.
Really not sure here though