I've always been disappointed when asking this same question, as I never get
any enthusiastic responses about the islands I occasionally visit (ie Aruba,
St. Thomas, St. Maarten, Grand Cayman, Puerto Rico) on business or pleasure.
When you get to these places and get on a bike, you'll understand why.

Not to burst your bubble, but cycling on many of these islands is not what
you'd expect. Mostly, when you get outside the small tourist areas, things
can get very "dark" very fast. I've never felt safe biking on any island,
with the possible exception of Aruba. Plus, unless you avoid the main
roads, you're often dealing with drunken tourist drivers - and in some
cases, drunken tourist drivers who are driving on the opposite side of the
road for the first time. I've had some very near misses on some of these
islands.

Lastly, the whole idea of "cycling along the beach as the sun goes down"
seems non-existent to me. Someone else may have had another experience, but
I've never seen those quiet stretches of roads or bike paths that pass right
by the ocean, as seen on TV and in travel guides. It seems all those areas
are eaten up by resorts.

I can't speak for Jamaica, Guadeloupe, or Antigua (roughly half the places
you mentioned), but I wouldn't build a bike trip around any of the other
islands above. They're all great places to visit, though. And once you get
to them, the pace is so laid back that you'll probably forget all about
wanting to bicycle anyway!