Yes, low voltage will cause a relay to not "pull in".
But I am confused by your discussion of 12 VDC and GFI's. GFI's are 120 VAC, not 12 VDC.
When on shore power, your inverter/charger should have you in the 13.2 VDC range (float voltage). Should be no problem IF the house bank is the source for signal to the relay.
But, if chassis battery is source and voltage is very low, certainly, that could be the issue.
SO, with a digital voltmeter, what are your readings at the chassis battery? house battery?