If you are plugged into 120 V, the converter will not come into the equation, anywhere, relevant to the residential fridge. You will be getting 120V directly from shore power, no battery or converter involved.
The only time the converter comes into play is when you are plugged in, or running a generator, and the converter is charging the battery and powering any 12V items.
If not plugged into shore power, you will need a battery bank and inverter large enough to supply 120V to the refrigerator. So what you need is a battery system large enough to power the fridge during your drive time. Whether or not your battery will do that, depends completely on the capacity of your battery, and the power demand of the fridge. So you have to crunch some numbers to figure that out.
In your shoes, I would get the fridge cold before you leave. Fill up some milk jugs 3/4 with water and freeze them at home, or some other form of block ice. Prechill the other contents of the fridge. Load it up and open it minimally. Maybe take a small cooler for drinks, etc. With this setup the fridge will not run very much and the demand on your battery while driving will be greatly reduced. When you plug in at night, stick the semi-melted ice containers back in the freezer, and move them to the fridge in daytime.