sam, I don't think your converter supplies any power to the furnace, it just charges the battery(s). The furnace fan is a DC device and it draws its power from the battery. Any AC powered device such as the Air Conditioner, Microwave, Refrigerator (if you're powering it that way), and water heater (again, if it's being powered electrically) are taking their power directly from the campground hookup, through a "bypass" around the converter.
I would not think your converter is bad from the symptoms you described, but your trailer batteries may be weak and, the converter may not be charging them if you're on a campground hookup. It is possible the 30A fuses in the converter battery charging output have blown. They are somewhat "hidden" in mine and I had to take the front cover off the converter to get at them. Check out my troubleshooting and repair in this thread:
https://www.hilotrailerforum.com/f28/converter-troubleshooting-repair-6454/
- Jack