Default should definitely be to shove here. I'd need a read that Villain was a really nitty opener (but also willing to call a shove with a reasonable range) to justify folding – you're right that calling from the SB is out of the question.

The thing about small pairs is that, assuming your opponent is opening a reasonable button range, there's nothing he can do to make shoving unprofitable. If he mostly calls with bigger pairs, then he's going to fold prettyy often, and you win so much from those folds that it doesn't matter that you're in bad shape when called. If he calls you more aggressively, then his calling range will consist of mostly unpaired cards, chich means that you'll be in pretty decent shape when called and won't need a lot of fold equity to show a profit.

BanTheRiver said:

Flatting is not the best idea, though with 27 BB's it's acceptable and done often enough. A better move would be to 3bet on the higher side. If he comes back over the top of you then you know you're either dominated or at best flipping and can easily fold. Considering that it's a button min raise it's very unlikely he has a hand that he'll be just calling your 3bet with; he'll either fold his steal hand or come over the top with 99+/AJs+.  Shipping 27 BB's just tells your opponent that you do NOT have either AA, KK, QQ or even AK. 

Hero has 22BB. Though even with 27BB, I would say 3-bet-folding is no good because you are going to have the right odds to call the shove. You aren't going to be “at best” flipping, you're going to be “usually” flipping, and with so much money in the pot you can't afford to fold away so much equity.

Even a conservative 4-bet jamming range for the button would be something like {88+,AJ+,ATs,KQs}. Note that that's about 8% of the deck, or 25% of a relatively tight button opening range. Your 4s have 38% equity.

If you make it 7BB (a large 3bet) and V shoves, there will be ~38BB in the pot, and you'd need to call 20 more. You'd need 34% to call. Given how close you are to having enough equity to call against a tight range, you'd definitely need to call against a slightly wider range. Better not to put yourself in that spot by just shoving yourself.