That baby is a citizen of the USA. Neither Howard nor Anna were citizens in the Bahamas. If Anna went for citizenship, why didn't Howard??....or did he??
Anna nor Stern went for citizenship in the Bahamas.
Anna (only) was given a Permanent Residency Permit but that is not Citizenship in the Bahamas. She got the Permanent Residency Permit because she claimed she had purchased a home for over $500K on the island. Stern is only a guest in the Bahamas right now.
Any child born to a non-citizen in the Bahamas is not automatically a citizen of the Bahamas. The child takes on the citizenship of the mother and father. Which are both American citizens. A child born in the Bahamas can apply for citizenship when they turn 18 years old but I would assume that the Bahama courts would consider if the child had lived their full 18 years of life in the Bahamas before they would give the child the Bahamian citizenship.