... I would think about some place other than Abaco. I thought you were talking spring/summer. The winter in the Abacos is likely to be just like Florida weather, you really need to be get into the Southern Bahamas before the weather starts getting nicer, the water warmer and less cold fronts making it there. Spring & summer weather in the Abacos is tough to beat. There can be nice weather in Marsh Harbour area in winter months, it will be ugly too, for a week or more, several times a winter season. Bring slippers and lots of fleece.
Georgetown weather is remarkably warmer and nicer than MH in winter, but they too can have crummy, cold days where you would not think of jumping in the water in February and March. The problem is where to leave a boat for extended periods in GT, and if you want to be plugged in, and I think that's what you said, I don't think there's a reasonable option. As Chris says, there are good private moorings available to hire, with all the protection you would want, but, you will need someone to watch the boat, run the engine, charge the batteries and you won't be plugged in. They don't call it Chicken Harbour for nothing ... Bombing around Elizabeth Harbour gets pretty limited after a week or so, then you are trekking back & forth to the Exumas, Long Island, and some of the Out Islands, and at this point, it probably would be good to have some Bahamas sailing experience under your belt, the cruising becomes more challenging, a certain level of experience is beneficial, weather will be a big factor, it can blow in the Bahamas, and in these cruising grounds all around anchoring protection is fairly limited to find outside of Georgetown. And if this fellow is in Staniel Cay for instance, and a strong front rolls through followed by several days of strong easterlies, he may find himself in a pickle to get back to the airport in G'town. You got all winter, fine, but on a schedule as tight as a week, it will get awkward.
I would agree with Blye, if you are talking about Dec thru Feb, you might consider being more south.
Cheers, John