I've been to Barbados four times after the hurricane season (mid October for about a 3-4 weeks each time) during the late 1980s and early 1990s.
The weather was always good and there was a steady temperature of 30c. As much as I liked the island as a beach holiday destination, I wouldn't want to cycle there as the roads were in a terrible state and the driving left a lot to be desired.
Go and enjoy the sun, rum, sea, beaches, cricket, calypso, reggae, food and wonderfully friendly and genuine people, but leave the bike at home.
Just to say that I have no idea how cycling would be on other islands in the Caribbean. And, who knows, in the 25 years since I was last there, things may have improved.