It s tricky... Even more since we didn't even go to Nassau at all but like everything in the Bahamas you need to be creative...
First we have a big cold plate freezer in the skylounge so we can pack most of the meat we need along with some of the seafood we can't get fresh locally (shrimp, salmon, etc).
Fruits and veggies are the big issue. We got some stuff in Gtown but selection and quality is limited. We were also able to get a few things in Staniel (tomatoes, salad, bananas, etc...) but you have to time it right depending on when the mail boat comes in.
Twice, guests flew in Gtown on a private jet so we were able to have someone buy fresh stuff in Miami, pack it in styrofoam coolers and bring them to the plane before departure
There is a new grocery delivery service in Nassau and they say they can ship to the islands, either on the mail boat or local flights. Haven't tried yet it but it is an option.. Also th grocery store in Staniel said they can order additional stuff and have it put on the mail boat.
Now, if you can go back to Nassau every once in while, it s lot easier. There is new supermarket right across Nassau yacht heaven which while not as good as they are stateside is still very nice by Bahamas standard. A little more $ than what we pay stateside but not to bad. In our case, we spent all the time in the southern exumas so Nassau was out.