I've used both types of fridges. The fridge in my first unit ran on
propane only, cooled reasonably well but not as quickly as the compressor models. I'm currently using a Dometic compressor model (can't remember the model #) with a separate fridge and freezer compartment.
I have 200 watts of solar and a 140 amp/hour deep cycle battery. I can boondock for months at a time without needing to charge my battery from anything other than solar. We use LEDs,
furnace, fridge, stereo, water pump and CO and gas detectors as much as we want. Battery has never dropped to lower than 75% full charge. By 10:00am the battery shows a full charge and stays fully charged all day.
Without solar, the same battery would last about 2 days before getting to 50% charge at which point I would switch to another battery.
If your solar is working properly and you have a good battery you should be able to boondock indefinitely as long as you have a reasonable amount of sunshine. If I winter camp I do take my
generator and actually used it once.
If your battery dies after one night, it is either defective or you have excessive drain.