Your 12 volt DC is not "completely" separate from your 120 volt system. You have a converter that changes the 120 volt to 12 volts DC.
I would start by locating the converter. Test the 12 volt system. It could be the converter causing the issue. It could be having issues with how the battery gets charged. Some older units have an internal relay.
It could also be losing 120 volts to the refer. Try resetting the rig's circuit breakers. Turn all off applying light pressure toward off. Then turn on main breaker, if equipped, followed by the other breakers one at a time.
Locate make and model of the converter.
Bud
USAF Retired
Pace Arrow
2003 Chev Ice Road Tracker