In an earlier rig I replaced our failed 3 way fridge with an 4.4 cubic foot electric fridge. The fridge cost about $120 at Lowes. I did run it off an Inverter. I will say it takes a 1200 watt Inverter. We bought the Inverter for $59 at a Black and Decker outlet. It's not that the fridge draws a lot of power but it takes a lot to start the compressor of the fridge.
Generally we did not run the AC fridge when towing. We also do not typically run our 3 way fridge while driving. Of course we only drive about 3-4 hours a day.
If it's really hot or we're going to drive a longer time, we ran the fridge on Inverter AC. On those occasions we keep the trailer's battery
charged as we drive from the tow vehicle. When we had solar
panels we kept it charged from the panels.
My recollection is that the battery
could run the fridge for about a day depending on the weather and battery without charging the battery.
We liked the AC fridge due to it's volume.
Of course there are some trailers that are AC only and AC fridges.
Hope this helps,