But it paternal paternity...is that even right...is proven, can she travel with Larry?
If Larry is proven to be the father and a birth certificate showing him as the legal parent can be issued by the Bahamian court - he would be able to take her to the US consulate and add her to his passport, or get one for her.
Since her name is presently Dannielyn Hope Marshall Stern -he may also want to change her name legally as well.
Children Born Abroad
http://travel.state.gov/family/family_issues/birth/birth_593.html