I don't think it would be a ground fault... I have a 3ft stake into the ground sheltered from the elements with a 6 ft wire connected directly to the charge controller and the inverter. Plus it runs fine from about 10:00 to dusk. Have I done enough to rule out ground fault?
Sorry I don't know the specs for your model.
If it doesn't have GFCI protection then it's definitely not that. Do you have an Amazon or Ebay link showing your exact model or a link to the manual?
If it's not a ground fault then it's faulting for low voltage as you have obviously been chasing. The voltage when battery voltage is higher makes it operate ok for the load but then it trips at night after you have heated the cabin a bit which increases the starting torque required and this increases amp draw etc and of course you no longer have 14.4 volts but maybe 12.8 or lower available.
Going to guess it's just an overly sensitive inverter due to it being cheap.
The inverter I chose could start a beam saw off less than a single group 27 battery even after it started to go below 12 volts.