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.
I guess it can work with 10A as well. I had one that melted in the plug that was connected to the original power out let at the back of the center console. But I think that plug itself was some cheap China shit.
I intend to use a relay that will be activated by the power to the old power outlet, that cable is now going to the USB charge ports but can also trigger the relay.
One easy way is to run a cable directly from the battery through the fire wall, and fused of course. In that case the fuse could sit inside of the relay box next to the battery.
Thanks for the suggestions, I'll report how it went.