If you have a charging line properly connected the battery should
receive charge when driving. Using a digital meter read the voltage at the battery with the connector to the car pluged in with the car running and when it is unplugged. There shoul d be abot 2.5-2.8 volts difference. If there is the battery may be bad and not taking a charge. How old is the battery and has it ever held a charge.
The American converter charges the battery through the same wires as it runs the lights
etc. I don't believe that they use a switch-over relay. Does it have a switch on the front that sez something like BATT-OFF-LINE
The converter should be charging the battery any time the rig is plugged into shore power. Make the same test as above, measure the voltage on the battery posts with and without the shore cord plugged in. It should go up at least 2.5 volts when plugged in.
The fuse that blew may be a reverse polarity fuse if the battery was connected backwards, if only for a second.
Read some voltages and let us know what you find. Be sure you are on DC VOLTS on a scale of less than 50 volts, 20 is better if you have it.