In order to answer your question, you will need to know the electrical specifications for the frig you plan to install. Unfortunately manufacturers don't typically publish the amp draw of their products. They do tell you average "KWhr" consumption for energy efficieny but that is useless when trying to answer your question.
I recently converted my RV to a residential frig. When I finished the install, I checked the amp draw on the frig and it was 9 amps when the compressor was running. That is the number you will need to know.
Inverters are rated in "watts". Watts is a product (i.e. multiplication) of Volts times amps. You know the voltage you have, 120v. If you can determine the amps, either by looking at the frig specs, or measuring it, or just guessing, you can calculate the watts. Mine was 9X120=1080 watts. If the inverter is serving the outlet for the frig exclusively, then a 1200 watt one would probably work. (Note that there will be a current "inrush" each time the frig starts, but that should work.) Most decent inverters have an automatic switch included that switches from the DC input to the 120 volt input automatically. You don't have to do anything to make it switch. Just connect the inverter to your batteries, and also connect the inverter 120V INPUT to your RV 120V system per the instructions provided with the inverter. When you plug into shore power, you will run the frig from that, and when you unplug, the inverter will automatically switch over to battery power. It is a seamless system.
The next question will be how big your battery bank needs to be and that will be determined by how long of a trip you intend to make before you plug in. The frig is not going to run continuously so maybe go by the energy efficiency label. I just looked an an LG that estimates 432KWhr per year. There are 365 days in a year X 24 hrs=8760 hrs in a year so the frig will run about 432/8760=5% of the time. If you are going to drive 6 hours, that means it will run 30 minutes (.5 hours) during the trip. (your numbers will vary) In 30 minutes, the inverter will use 9X120X.5=.54KWhr during the trip. Your batteries will have an "amp/hour" rating. This rating tells you how many amps you can draw from the battery (typicaly for 20 hours) at a useable voltage. It is not good to totally discharge a battery. How long can you run a 9 amp frig on a 150 amp/hr rated battery before it goes to 50% is the real question. Divide 150/9=16 hours. Since our example says you will operate .5 hours, you should be ok. Notice I did not include any inefficiency in the overall system. 2 batteries in parallel will give you twice the capacity also.
I have no doubt my comments will solicit further comments. Good luck with your project.