On early years of flotillas in the Med, with no fridge on board, I took a netting shopping bag on purpose to dangle beer cans in the sea for use when a shore-side bar was available. But did have to defend against swimmers from the rest of the flotilla...
Unless your fridge is plumbed-in such that it dumps the extracted heat overboard (and not into the cabin) it is not going to operate as air-con at all!
Probably better to use bought ice in a freezing mixture to freeze bottled water (assuming you can trust that...). We used to buy block ice intended for fishermen and break it up in the cockpit bare-chested to get full benefit...