When the shore power was interrupted , the inverter may have been overloaded and blown it's 12 volt power supply fuse or tripped the breakers on the side of the inverter .
Do you have info on the inverter ? Brand and model ?
Do you have a voltmeter?
Inverter 12 volt supply fuse will be within 3' of the coach batteries and probably look like the photo below.
99DSDP 3884, Freightliner, XC, CAT 3126B, 300 HP /ALLISON 3060
2000 Caravan toad, Remco & Blue Ox.