By the way, what you would need to do if you travel on just an ID/BC is get an emergency passport from the US embassy/consulate in the country you are suck in. They might issue a passport with just the documents you have (BC) or they might issue a one time entry into the US. They aren't going to make you live in Nassau forever.
People have needed emergency passports before, and the US official know how to do it, but it is just not a problem you want to deal with on top of the emergency. Also, some places don't have an office or official, and you'd need to get it from another island, so that would be another delay.
My sister had her passport stolen in France once, and she had to spend a day getting a new one, involving a few trips back to the embassy.
Nancy