Yes and Yes! I think you answered your own question there? September is the worst month you could possibly go! Although the ship might be able to stear around bad weather , it is no fun visiting Islands that have just been flattened! Anyway, most Cruise ships do not move from the Med to the Caribbean until November, so your choice of cruises would be very limited. Happy cruising.
Officially, hurricane season stretches from June 1st to November 30th, and hits not only the Caribbean islands but also parts of Mexico, Florida and the Texas coast. Within that time frame, August to October are peak months. And within this peak hurricane season, early to mid-September is the pinnacle.
Cruise ships are a practical way to visit the Caribbean during the risky season, as they can simply re-route if a storm threatens. Choose low-risk spots -
although there's not really a totally safe zone, the southern ABC islands -- Aruba, Bonaire, and Curacao-- are considered the best bets. Personally - as its a honeymoon I would avoid the caribbean or southern states/mexico and go to the Indian Ocean ie. seychelles, maldives etc. Whatever you do have a lovely time and best wishes for the future.