Having trouble dealing with postflop minraises

[KCW12]

A particular situation I struggle with a lot is getting away from a strong but vulnerable hand when I get minraised after the flop where calling would end up committing me to the pot. I don't want to dig through my HH right now, but I'll make up a hypothetical hand that I would have this problem with:

$22 sng, 4 players left, stacks are:
Hero: 3000
Button: 3000
SB: 4000
BB: 4000

Blinds are 150/75, I get AJ and raise to 400. Button calls, both blinds fold. The pot is now 1025. The flop comes AQ9 rainbow. I bet out 600, villain minraises to 1200. Villain is an average $22 player, and the minraise is a strange play that I haven't seen him make yet. Now what do I do?

Just calling would pot-commit me, as there would be 3625 in the pot and my stack would be down to 1400. So calling is out.

If I reraise all-in, I likely have very little fold equity, and I just have to hope he's making some strange play with a draw or something.

Can I really fold this? I have top pair with decent kicker, and had villain made a larger raise, I probably would push.

In these situations, I can never convince myself to fold, and I push all-in and am almost always called by a stronger hand. I know the losses probably stick out in my mind more often, but it really seems like a minraise here is almost always a big hand, yet I can't ever fold in these situations.

And I know some players would play differently preflop or first to act on the flop, but let's focus on the minraise part. I've run into these situations a lot when I'm the BB as well and it gets limped and I hit a decent hand like a TP/draw combo and get minraised as well.