The baby IS a US citizen, unless there have been some very recent changes to Bahamian citizenship laws. By virtue of being born there, she has the right to register as a Bahamian citizen once she's turned 18, but has no Bahamian citizenship now (except that I heard there's now a third guy claiming to be the father, and if he's a Bahamian citizen, then the baby's Bahamian citizenship status is undetermined, but her US citizenship status is still secure).
Oh , OK, I had no idea that in the Bahama's you can have US citizenship when born there. I thought you had to be born on US soil for that privilege.