The answer is that you should never have an M of 3 (unless you have a slightly shorter stack all-in and he wins). You should start shoving all-in instead of a normal 3x raise when you get down much below an M of 10. Some say even at 10 or 9, but I make some allowance for turbos, because everyone has that low an M and you're going to get called liberally.
You're going to get called by almost any hand when you have an M of 3, because you can't be giving much worse than 2:1 pot odds
to the BB. If you should get down to 3, then you should either shove with any two and hope to get lucky, or you should try to pick up a hand before the BB hits you (if you have that luxury).
The hands you've listed (A2, K4, Q7) are some of the worst hands to be raising with in a 6-10 M situation. Very rarely will they be in ok shape versus a call (maybe Q7 on a fluke versus an underpair). When you're down to 5 or lower M, then A2 gains some actual value, because you might be called by KQ, KJ, QJ, etc. My advice would be to pick hands like suited connectors (or one gappers), any pair, and any two high cards. If need be, throw in some steals with any two cards, just to keep your head afloat by taking down the blinds. Don't be afraid to get it in with a bad hand. To anyone who really knows, it will not reflect badly on you as a player, because it's a mathematically correct play (due to the size of the blinds and the high reward:risk ratio).
So in short, I would advise shoving instead of making a normal raise with a short stack, and I would play A2o, etc. the same as I play 72o (i.e. it's only good for a total steal). If by any chance you do get to an ultra-short stack, then I'd open up to include any A, or potentially any two cards.