As you will have discovered, there are three seasons for Caribbean villas--low season, high season, and Christmas season. Your budget of $1200 per night will work, but your planned six night stay may leave you one night short of the week required by many villas. Some will rent only for a full two week period over the holidays, and others won't split New Year's from Christmas. That said, there are always exceptions.
"Easy access to a beach" would rule out most of the great villas on St John and St Barth, which are perched on hillsides. You must jump into your jeep to get to the beach, of which you will have many from which to choose. For great villas right on the beach look to the Terres Basse section of French St Martin, and to neighboring Anguilla. Also Turks & Caicos and Barbados.
Direct flights are also a problem, since DC is not a major hub for Caribbean flights. If you're OK with "on the beach the same afternoon" then most of the Caribbean opens up for you.
On most of the current and former British islands your villa will come complete with a staff. If you like the idea of a cook standing by in the kitchen, waiting for you to order something, that's great. If you don't, you may look more toward the French islands or St John, where you typically will receive once-a-day housekeeping/pool service, but won't have anyone underfoot.
Wimco has more villas than anyone else, and their website will give you a good idea of what's available, and what it costs. You may or may not receive good, personalized service from Wimco. If you decide to investigate St Barth, I strongly recommend
St Barth Properties--they will treat you very well, and work hard to find you exactly what you want within your budget. As a party of 6 you are a perfect candidate to save money by chartering a flight from SXM to SBH.
One option you may consider is the combination villa/hotel, exemplified by
Green Cay Villas on French St Martin. Here you will have all of the privacy of your own villa with pool, together with the services you'd expect from a fine hotel's front desk.
Enjoy the Caribbean.