I'm sorry. That is awful. I honestly can't understand some people. I've alway thought "better safe than sorry." With any luck there will be no problem with it just being the bahamas. The Bahamas, unlike many other destinations does allow admittance for U.S. citizens with just a birth certificate instead of a passport (so I suppose you can assume they are more liberal).
If you do intend to make a last ditch effort to convince the child's mother that its necessary (especially to give you an even better piece of mind) you could refer her to this link from the US State Department: http://travel.state.gov/bahamas.html
Under entry requirements for the Bahamas it states the following: "In an effort to prevent international child abduction, many governments have initiated procedures at entry/exit points. These often include requiring documentary evidence of relationship and permission for the child's travel from the parent(s) or legal guardian not present. Having such documentation on hand, even if not required, may facilitate entry and departure."
Be advised that this advice is given for most countries and its definitely not an absolute so in any event try not to worry because I'm sure you'll all have a wonderful time.