I appreciate all the replies! We talked with the manager of the storage place and he said that they hadn't had any issues in our unit prior, but they didn't have an RV in there, just a bunch of household stuff (I guess they were moving?)
They did agree to have their electrician come out and test the GFCI to see if it was overly sensitive/failing and change it out if needed. I'm hoping that helps.
The batteries seem to be holding up well, looks like we'd get about 24 hours on them with the residential fridge running. So I guess that's the bright side.