From what you have posted, here is how I would proceed:
(1) Since everything works OK with the 12vdc battery
, I would assume the 12vdc circuits and loads/accessories are OK.
(2) Since the main breaker (assume this is the 110vac main breaker?) pops out, I would be looking for and trying to isolate a problem in the 110vac circuits. One possibility is that the main breaker itself needs to be replaced. Another possibility is that there is a problem with the 110vac-to-12vdc converter. Measure the current drawn at the main breaker. If the current drawn exceeds the main breaker rating, then you know the main breaker is OK. If the converter is the ONLY 110vac load that is turned on at that point, then I would suspect a problem with it. If the current drawn is below the main breaker rating, then the main breaker needs to be replaced.