Your "inverter" tripped out because of the 240 volts applied. It could be history after you get the transfer switch replaced.
Your energy management system should be OK.
You may need other items as well. Converter could be fried, microwave and anything else 120 volts AC that was on when the 240 volts was applied.
You must have 12 volts DC for your furnace to operate. I am guessing that the converter is toast and the batteries are run down. You could try starting the rig, then try to operate the furnace. While the rig is running, 12 volts DC should be supplied to the rig. Lights and other 12 volt DC items should work.
If your lights are currently working then there is another issue with the furnace. Maybe a blown fuse.
I am guessing that your rig is 50 amp. Is this correct or is your rig 30 amp?
Bud
USAF Retired
Pace Arrow
2003 Chev Ice Road Tracker