I know that some use that type of system with success with solar to recharge the batteries. But my opinion is that you want a gas fridge for boondocking. I believe a residential fridge compressor draws from 1 to 3 amps while on, with a startup surge that is briefly higher than that. To provide that AC power via an inverter from a 12V system, you can guestimate that the DC amp draw would be AC amps x10. So 10 to 30 amps from the batteries with a startup surge of something like double that. Now there's duty cycle to consider since the compressor only operates as needed, but bottom line is that is no small amount of DC capacity you'll need to have to make sure your fridge keeps running. If you assume 50% duty cycle with the low end of amp draw, that's 12 hours x 10 amps for 120 Ah per day. Following the rule of thumb of avoiding drawing your batteries below 50% capacity, that means 240+ Ah capacity per day JUST for fridge. You'll need DC for other stuff, too, so that's even more battery capacity required. Unless you have a humongous battery bank, you'll need to fully recharge those batteries daily or you're out of power after 2 days.
A gas absorption fridge burns propane, I'm not sure how much but it isn't a lot, and has a continuous DC draw of about 1 amp to power the control panel. My opinion is that is the better option for boondocking refrigeration.
2012 Aspen Trail 2710BH | 280 watts of solar on the roof | 2x6V GC batteries | 100% LED lighting | 1500 & 300 watt PSW inverters | so far strictly boondocking