Are you able to bring a birth certificate?
Most cruise lines typically allow passage with a state id and birth certificate. For example from the Carnival web site:
US-born citizens may also use a birth certificate (issued by a government agency) and accompanied by government-issued photo ID; a Mobile/Digital version of a photo ID is not an acceptable form of identification.
This is for cruises that begin and end in the US.
This policy also applies for MSC Cruises.
Typically, once you are accepted on the cruise ship no additional clearing of customs is necessary in the visited ports. So if this cruise line allows you aboard, they will not further check your credentials in the Bahamas.
Please check with the specific cruise ship for details.