I mostly boondock and would never
consider anything other than a propane fridge. I have 200 watts of solar which helps but will not keep my 2 batteries fully charged when I use my Direct TV for any length of time off the inverter.
I highly recommend you get a little portable inverter generator. I use a Ryobi 2200 watt unit which I absolutely love. It will run my converter to charge my batteries without any effort at all. It will run all day on a gallon of gas.
I just arrived home from a 3 day trip where I used that generator every day. I set my propane fridge to run off propane rather than 120 volts which allows the generator to throttle way down because it is basically just running the converter. Generator is then fairly quiet and very fuel efficient. So much better than my big 4k generator. I couldn't be happier.
I have a voltage/current meter that I constantly watch to know the status of my batteries. When I turn the inverter on to watch TV at night I see it pulling almost 10 amps from the batteries. Last night the batteries started at 12.7 volts when I had shut down the little generator. I watched it slowly creep down and after a few hours the batteries were down to 12.4 volts. Not a big deal as I would let it go down to 12.1 volts before I get concerned and would shut things down. This drop was from 2 brand new Costco deep cycle batteries.
Inverters suck battery power, period. A quick rule of thumb is that the 120 volt current that is required will equate to about 10 times the amount of 12 volt current required. You can't pull power from thin air. Output power for 3 amps at 120 volts would equal 360 watts. Input 12 volt power for 360 watts would be 12 volts times 30 amps = 360 watts. Pulling 30 amps continuously from batteries will drain them in no time.
In my opinion as an electronic technician, you will never be happy with a couple of batteries and some solar for a residential fridge. Remember, moon light and hazy days will not produce any solar power. Once the batteries are dead, so is your food.