15amps??? Is it a compressor fridge or a cool-box,, the only time them cool-boxes draw 15amps is when you set them to heat, the standard fag lighter in the back are sufficient for a cool- box, if it's a compressor fridge, then I would be looking to run direct from battery.
The Peltier cooler ones pull big currents, and most are Peltier ones now as cheap and easy to make.
My convection fridge I use for camping - the type that can run from 12v, 240v or gas - is about 60W on electric, so around 5A, and runnable off a ciggy socket.
15A is probably beyond most ciggy plugs (a quick squint around the electronics retailers seem to put them around 10A max, and beyond that they are liable to heat up and soften), though I notice the OP is using a proprietary plug.
Personally, I think, for a constant 15A, I would run a separate supply through one of the 80A fused supplies, with an (accessible) inline 15A fuse, and through a relay thats energised when the car is running only.... ....look at HRW circuit for how this is achieved.