I'm looking for a large-scale geospatial data set (or several components that could be merged) that includes all the islands of the Caribbean region. The trick is that as many of the physical individual islands as possible need to be listed individual features in the data table. In other words, I want the data set up so that it is easy to select particular islands (not countries) and style them individually.
I've tried Open Street Map -- specifically Quick OSM and the Overpass API, using the place:island tag. That works pretty well, but there are a not-insignificant number of islands missing (e.g. most of the bigger Florida Keys, the main island of Cuba (though the smaller islands of Cuba are included), some of the main islands of the Bahamas, and others).
Natural Earth has two problems -- the scale is not small enough, and the each table row represents a country (group of islands, or mainland and islands together), rather than an individual island.
Any suggestions much appreciated.