I agree with 3for3’s reply that (a) the very unlucky/unlikely result of this hand shouldn’t be given much weight and (b) just shoving preflop is probably preferable, hoping to get into a coinflip with dead money/overlay, though if the players left to act are good, they may realize that you can be shoving pretty wide and adjust their calling ranges accordingly. Flatting to set-mine is less appealing than it seems for the same reason that bet-sizing post-flop is tricky: Once you flop your set and bet into a dry side-pot, you cannot rationally be bluffing, so your opponents have less incentive to call than they would normally. With a nutted hand you gain very little from fold equity by betting the flop, and because your opponents know this you are unlikely to get much action from anything worse than exactly AJ. That’s why I slightly disagree with 3for3’s bet-sizing thoughts … If I’m in the BB with A9 and I think hero is not a maniac/idiot, I’m probably folding to a 2/3-pot bet. So I think your small flop bet makes sense, and I think villain’s call with 2nd pair is pretty bad. Having said all that, if your opponents are really bad then all this goes out the window and you might as well just bet half pot on each street and hope villain just calls along with Ax.

I would be really interested to know how a solver handles this with all players acting more or less GTO. Sadly I don’t have flop-solver access/skills…