I have the residential fridge and 1000 watt inverter. Rather than run my battery down while on the road I freeze water bottles in the freezer. When we are ready to leave I put some in the fridge and leave some in the freezer. The ice keeps the fridge cold all day. The ones in the freezer never even thaw out.
If you have something that needs to be kept extra cold put it in a plastic bag along with the frozen water bottles. The bag will keep the cold next to the food.
I have quite a bit of experience with sailboats. They install 'cold plate' refrigerators in yachts - basically the compressor freezes a cold plate that is filled with liquid and the frozen plates hold the temps in the fridge for 24 hours in tropical heat. Same concept freezing water bottles in my residential fridge.
If you want to run the inverter off the truck consider using one of your upfitter switches to power an extra line to the trailer battery via a charge controller. My F250 has 2 25 amp upfitter positions.
There is a guy on youtube who mainly boondocks and runs everything off electric. He's got some big batteries and he started out by carrying some portable solar panels that he set up by his rv worked great and he could max them by pointing towards the sun during the day. Later he went to panels on the roof but if you want to keep from having a huge battery bank add enough solar to run the fridge during the day and you'll only need a night's worth of battery. Forgive the over the top cute of the video Portable Power – Solar more connected than Kevin Bacon