Since you are building from scratch, you have the advantage of being able to double the insulation of the fridge by insulating the framing that holds it. Use polyisocyanurate panels (available at Lowe's/Home Depot) and seal all gaps with foil tape. PIC is better than styrofoam or polystyrene and nearly impervious to moisture.
The "startup" cooling issue you mention can be mitigated by using a block of ice (or multiple ice packs, which allows more strategic packing) at the beginning of a trip.
Also, be sure to install silent
12v fans for venting. I initially used some computer fans I had lying around, but eventually swapped them out for some quieter computer fans, as the "average" ones were much louder than I expected, especially at night. It's good to put a thermostat on the fans, but I also installed a switch, so I can turn them off manually when the fridge is not in use, and the ambient temperature is high.
I know this information doesn't help with "which fridge," but the information can be applied to any and all.
I think technology is getting to a place where *just* electric can be a realistic option, even when boondocking (especially with solar), but I still think having the propane option is best if you plan on doing even a modest amount of boondocking.
Also, I would *NOT* go with a regular apartment/office/dorm fridge. Build quality is low on those to begin with, and the problems are only compounded when you put them in something subjected to huge temperature fluctuations and vibration.
You also mention having to level the fridge, which is a very real thing. I've been surprised by how finicky those sensors can be. I considered building an internal leveling system for just the fridge, but ultimately, especially with such a small trailer, I think it's best to just get the rig leveled properly when you camp.
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