Why would you ask if you can get off the ship in Castaway Cay if you can't in Nassau? They are both under the jurisdiction of the Bahamas. It's kind of like asking "Hey, the USA doesn't want to admit me -- but if I only go to Disney World can I get let through customs?"
You also can't board the ship with invalid Bahamas' documentation by claiming you just won't get off the ship. Either you have your documents, or you can't sail.
In terms of birth certificates, be careful. Even if Disney and the Bahamas accept them, realize that if the worst happens you will be unable to FLY home from the Bahamas without a passport if you get sick / ill while on the ship or in port. The last thing you may want is for an unexpected heart attack or other major medical issue to hit and then being told you can't be airlifted home because of your lack of US citizenship documentation. And a birth certificate is NOT good enough for air transport.
You also can't board the ship with invalid Bahamas' documentation by claiming you just won't get off the ship. Either you have your documents, or you can't sail.
In terms of birth certificates, be careful. Even if Disney and the Bahamas accept them, realize that if the worst happens you will be unable to FLY home from the Bahamas without a passport if you get sick / ill while on the ship or in port. The last thing you may want is for an unexpected heart attack or other major medical issue to hit and then being told you can't be airlifted home because of your lack of US citizenship documentation. And a birth certificate is NOT good enough for air transport.