Sam, yes lots of airports here in the region - each island has at least one, but not small strips, most are quite long - Bequia is around 3,500ft (it has an "interesting approach"), Union Island is short and had a very, very interesting approach. Most islands one can land an wide bodied jet St. Vincent, Barbados, Tobago, Trinidad, Grenada, St. Lucia, Martinique (FdeF), Canouan etc within an hour and a half of here. Mustique need special training like St Barths. Dominica has an interesting approach on one of the airports. Guadaloupe and Martinique, the French islands, have lots of small airports. St.Kitts and Nevis have one each, Nevis is quite short, Antigua long runway bumps approach - SLN and Antigua are about two and a half hours way in my PA28.
Yes, we have brilliant beaches here in Barbados and in the region generally.
I always thought a rally around the islands would be fun but it is a long way from anywhere over 1,500 miles from Miami so not close at all - usually take 20+ hours of flight time to get down here - I did it is under 15 a year ago but I took a slightly more adventurous route.
Was thinks for the golf nuts - a single club golf tournament - one hole per airport (seems to be a gold course near most) would be interesting. Flying around down here is fun and even in hurricane season the weather is good - from end of November to late May near on perfect.
If you're thinking of organizing something let me know how I can help.