Whew, okay, I'm glad I'm not the only one confused.
If your battery is getting charged when you're plugged in at home, but not when plugged in anywhere else, well, that's just downright confusing. I wouldn't even know where to begin with that, if it's even possible...
The fridge is likely a 2-way absorption fridge, which uses a 120V heating element when plugged into shore power, and propane for heat when not plugged into shore power. It always uses 12V to run the "electronics".
You said that you measured your battery at 7V (or something much less than 12V), which indicates you do own a multi-meter. You need to plug the rig into shore power, and measure the voltage across the battery posts. Write that voltage down. Then go unplug your camper, and wait about 5 minutes, and take the battery measurement again, and write that number down. Then fire up your generator (assuming you can), wait about 5 minutes, and measure at the battery again.
Good readings at the battery would be:
Plugged into shore power or generator running: 13.2-14.4V
Unplugged from shore power: 12.2-12.7V
If those are the readings you're getting, see below for the potential for enough voltage drop in the 12V fridge wiring to cause it to give a "low voltage" alarm.
If you're getting anything less than 13V at the battery when plugged in to shore power or generator running, then you have an issue between the converter and the battery. Could be a loose wire, could be a blown fuse, could even be a bad converter. Next step in that case would be to go back to the converter and take voltage measurements in and out; you should be getting 120V into the converter, and 13.2-14.4V out of the converter. If you're getting less than 120V into the converter, then you need to continue up the chain there (there should be a wire going into a standard 120V outlet, than feeds from a slot in your AC breaker panel). If you're getting less than 13.2V out of the converter, then you need to check the fuses on the converter. If the fuses are good, but you're still not getting 13.2-14.4 out, then you might be looking at a bad converter.
If you're getting anything less than 12.0V when unplugged, after you've been plugged in for a while with the converter charging the battery (which you verified by taking a measurement at the battery while plugged in), then you likely have a battery issue.
Now, all of that said, there is a situation where you can be running in "propane" mode, and the fridge is not getting enough volts. But IIRC that happens regardless what your power source is (converter or battery), and it's because the wire gauge that was used for constant 12V to the fridge is not appropriate for the length of the run. So you'll get such a drop in voltage that the fridge won't get enough, and it'll show a "low voltage" error, even though each source may have plenty of volts & amps to dole out.
Only you can gauge your comfort level with actually conducting repairs, but if I were you, I'd stay out of the 120V AC system. If you find a problem there, probably best to hire a professional. But the 12V DC system is not quite as dangerous until you start replacing wires and making connections. A wire not gauged properly, or a loose connection can cause things to melt and catch fire. Which can happen in the AC side as well, but that side can also kill you pretty quick...
