I think I would shove Chris. With half of our stack in, a 7 blocker which is somewhat relevant, and the ability to deny any kind of free cards I think I would just presume shoving is better than checking. Not a spot we are thrilled about of course, but I do think it's better than checking, especially if we are planning to check/call at a high frequency. Betting will just be better in that case IMO.

I think that flatting badugi AA combos (especially AA72, yuck) preflop should be part of folks' strategies, but I don't think this is the spot for it. I think you want to discourage BB coming along and encourage BTN to 4bet with worse if they're as splashy as you suggest (though some otherwise splashy players don't include 4betting in their splashiness).

Even more generally, I don't really like calling the SB to a BTN open. I'm typically either folding or 3betting for both isolation and value (as we should be pushing equity versus a BTN open range by playing pretty tight in the SB, imo).

Beyond that, I completely agree with Nick Johnson . It's pretty hard to be making too much of a mistake (versus BTN's range) by shoving here. Sometimes you'll be drawing basically dead, but that's only sometimes.

Preflop, in a standard game with full stacks (100bb) decent players, this would be a clear flat preflop for me.
You have kind of the worst AAxx possible, and are in the worst position.

But here, as you described Villains, and as BB is short stack, I would 3bet this to get action.
Except if you have the read "BB aggro vs opr, but sticky vs 3bets". Here is ok to flat and let BB enjoy his move. But it's a very specific scenario

Flop, with this SPR you can't go anyway but get all in
actually, this is not such a bad flop for your hand (it's rainbow, the connection flop isn't low (69T would be a worse flop for exemple)