I don't have my Bahamas Pilot's Guide with me, but I don't think there is anywhere else in the Exumas big enough for you. How long is the Norman's Cay airstrip? It is uncontrolled. I might have a slight worry over leaving a plane there unattended, probably unfounded. You might also have a look at the Stella Maris strip (4,000 ft) on Long Island. Again, if it gets windy you could be stuck there, but it's quite nice to explore the island.
My pick would be Nassau over Georgetown. You've got sheltered water all the way South over the Yellow Bank to most of the Exumas. The trouble with Georgetown is the last 20-30 miles over Exuma Sound which is fairly open. If it's windy you can get stuck in Georgetown Harbour. There are worse places to be.
We have been to the Emerald Bay marina on Great Exuma, back when it was a Four Seasons Resort. After going broke it is now Sandals. This is a couples-only resort and I got the impression children were unwanted, so have not been back. The floating docks are very nice. It's open to the east, however, and I worry about surge when seas are just right (or wrong). I would be afraid to store my boat there unattended.