The 380 is about the smallest cartridge recommended for primary self defense usage. If one uses good, made for the purpose self defense rounds it should be a decent performer. The Browning BDA is a good pistol though not as popular as it could have been. I think it was another made by SIG for Browning but I am not sure without looking it up.
I agree - I personally wouldn't go any lower than 380. But with it make sure you get some good performing ammo - oh and make sure to practice some with that ammo not just the cheap stuff. You don't want to find out your gun doesn't like Brand X or isn't as accurate with it in a SHTF senario when you're counting on it to perform.