Well AQ is a hand that mostly makes top pair. If my opponent & I are both 200bb's deep, I start losing enthusiasm for stacking AQ in a 3-bet pot when I hit top pair, since for villain to put a 200bb stack in they typically have better in a 3-bet pot.
Think of it this way: If you open to 3.5bb's, villain 3-bets to 11bb's, that puts 22ish bb's in the pot, with us having 189bb's behind. So in a 3-bet pot with double stacks, the SPR is still 8.6! If you look up
SPR's you'll see that they list hands like top two, sets, and good draws as hands well suited for this type of SPR.
Since we can't get the SPR right, I typically just flat AQs here. We keep the pot small & our opponent's range wide for the times that we make a pair, and we have tons of implied odds for when we make flushes/straights. And we don't have to worry about being stuck in an in between SPR.
So oddly enough, I think you've been doing it backwards. When you're 100 big blinds deep, I 3-bet AQ more often. Because when I make top pair in a 3-bet pot, the SPR is about 4. When I'm deep, I flat more often.
And a logical extension of this is 3-betting implied odds hands when you're deep. I 3-bet small pairs and suited connectors like crazy (and I 3-bet very small) when I'm deep stacked. If villain raises 3bb's, and we 3-bet to 8bb's, we're often left with a great SPR for implied odds hands, but our opponent will still stack off with top pair & over pairs.