What type of boat are you renting? The smallest catamarans from Moorings/Sunsail have a generous drawer freezer that stays very cold. I'm sure its cold enough to make ice, but not a practical way to keep a tray of water in there. We fit two 10 pound bags and some other crap into ours. IIRC Moorings gave us four bags of ice free; you can buy more at the dock before you leave if you want it. They also gave us two pretty good large coolers that will hold ice for a few days. We left one cooler ashore and used the other to precool the beverages with the other two bags of ice.

Some boats have a second dorm fridge which is great for cooling down bevvies before making drinks to conserve ice...and who wants melted ice diluting the alcohol anyway??? Our second fridge was broken though, so we rotated beverages from the cooler into the refrigerator as room allowed. We were able to make our initial ice provision last all week, but if you need it, it'll be available for a premium price at Staniel Cay and Highbourne Cay. Normans doesn't have a store, but if you eat at the restaurant, which I highly recommend you do, you can beg some off of them too.