I doubt it will be a problem. We’ve cruised extensively in the Caribbean and we did the best we could but some weren’t on the list probably because they’re territories of other countries. It’s not like you spend days on an island during a cruise and you certainly don’t go through any kind of formal immigration/entrance process with island officials. They may board the ship and glance at passenger lists but I’ve never had them ask to see or handle my passport other than at initial embarkation.