Spring in general, April, May and June I think are the best months. Good weather, and a great overlap of many species of fish to catch.
I use to keep the boat at Atlantis/Hurricane Hole in Paradise Island every Spring because it was an easy Hub to fly in and out of with family and friends. I like it there because depending on the wind we would fish off Chub in the pocket, Andros, Hole In The Wall, or run down the Exuma Bank and fish the Exuma Sound.
I don't care one way or another for Atlantis because it's a little to commercialized BUT the kids LOVE it and its a nice Central location to fish multiple Bahama Hot Spots depending on weather. In the spring if you get a South West Wind along with a change of the tide the fishing in the Exuma Sound up in the North West Pocket can get Red Hot. Typically there is little to no wave action and in one day you can hook up with Marlin, Sails, Wahoo and a boat load of dolphin with 4 to 6 hooked up at a time. Of all the places I've been, the Spring Fishing in the Exuma Sound is my favorite. Calm seas, great fishing, crystal clear water and tons of places to dive, and explore on or near the bank. Just a beautiful place to be and only a 1 1/2 to 2 hour run south of Nassau.
Fish the day, Dinner, Casino and Luxury Hotel rooms at night. Like I said I personally could care less for Atlantis, but for guests it's great. At sunrise they show up at the boat we fish, swim, and Island hop all day long, then a 2 hour ride and a nap later they're back at the Hotel.
Don't get me wrong the Abacos and Harbor Island are beautiful as well, but the Spring fishing and diving in the Protected Waters of the Exuma sound are second to none for me.