Look into private cat charters in the BVI's - ask Dr. Lynne (TS&M)
All the BVI's are close enough to be reached in one day or less and you can go over the the USVI's also. Some of the boats have compressors, if not there are options for getting tanks at the diveops on each island. Several diveops on Virgin Gorda (DiveBVI) and Tortola (Blue Water) do rendezvous diving - they meet you at your mooring with tanks/DM etc. - at just about any of the smaller islands in that area.
Snorkel the Baths off VG, dive the Rhone, the Indians, the Dogs, Norman, Anegada and many more good to great divesites. Shop in St. Thomas, dine on St John, drinks at Pusser's, Foxy's, the Willy T - all the various smaller islands have dockside bars - it's all possible within a week. In that area a lot of the amenities are clustered around the various ports since it's a sail-driven economy so you typically walk {or short cab ride} to nearby stuff.