I guess it is the electrical engineer in me that makes me want to reply. First, the inverter has many useful purposes but running a compressor is not one of them. A Compressor has a starting load and depends on a exact sine wave uniform operating signal. Though,inverters are getting better at this, the first bunch were really were only square wave generators. They are not suitable for running motors and will quickly burn them up. Getting a high current sine wave inverter that will run a compressor is very expensive and not very practical. The next problem is finding a battery which can supply the current needed to run a fridge and the inverter. If you need 5 Amps continuious at 120VAC then that means you need about 50 amps continuious at 14VDC. You can imagine how long your battery will carry that load on a charge. An AC generator produces a good sine wave and is a lot cheaper. Even bypassing your problem of 50 Hz which should not be overlooked, the real solution is a LP refrigerator which uses about the same amount of energy to do the same thing. It is quite convenient for MH and Campers though. I know that this is not what you want to hear. Ice is also a good energy storage source and people who want to avoid LP camping in this country choose to plug into the mains at campgrounds. My engineering jobs took me through a stint with Gas refrigeration, other types of refrigeration, and solar. No easy answer yet.