Pre: fold to the 4bet. Yeah, it's click-y, but 4bets on the whole don't have to be big because the range is so condensed. Devil's advocate, you're on the BTN, you're deep, it's basically a click-back, and flops are fun. Fwiw, I don't have much a 5betting range unless I know villain has some 4bet bluffs, so I'd be flatting plenty of my 3bets (though I don't expect villains to know that) and I'm happy to cut bait with these marginal 3bet bluffs. But calling is fun. Not gonna knock it too much.
Flop: I actually like just flatting here. I'm not really worried about getting overflushed (some AhXh or KhXh combos, c'est la vie), more so that our 3 combos of 99 are our only strong hands and there's a flush draw staring villain in the face that will be easy to continue against with his overpairs and nut flush draws. I'm usually just flatting 44 and 33 on the BTN pre, though maybe you're 3betting those some, too. I do think most villains will also suspect slow-played AA from us, at some frequency, so six more there. I think a flop raise is totally fine, because it's not really a bad event if we get 3bet unless we're up against those better flush draws. But the hands that we're targeting for folds, we are crushing equity-wise and can target on the turn. I'd delay aggression. With our AA - TT holdings, are we just flatting, or is there merit to a flop raise for equity denial? My default is just to call, but maybe I should look at that default more closely.
Turn: Unfortunate card for our bluffs. Do you ever have Ad9d here? I'd guess no, and it's hard for me to find other 9x in range. The flop raise means we constricted villain's range, so he's only got overpairs and better flush draws. I don't think any one of those hands folds here on a very good turn blocker. With very little fold equity and only a jam making sense with the SPR, I'd probably check behind and see a free river. We've still got good equity when called (except by AhKh, AhQh, etc.), and if you think we can get QQ, JJ, or TT to fold to a jam, there's merit to it (I think AA and KK are snapping, especially with no heart). Again, with a front door flush draw staring villain in the face and our likeliest value (99) being reduced to one combo, I think villain's overpairs are usually calling.
All told, I think the small turn bet is the only truly weird play? Definitely priced in to call, and glad you got there!
A related question for you that this hand made me think about: how are you playing your overpairs in this situation? Whatever mix of AA - TT you'd arrive at the flop with.