Check the battery voltage at the terminals with the heater running. Should be around 13.5..
If the converter can't keep up the voltage will be down. Not all converters can run without a battery but yours might.
This is not complicated. Battery charge voltage should stay up with the furnace running. If it does not and your battery is old, replace it and begin over. Batteries are not purchased, they are just rented.
Sure you can test the voltage from the converter with it disconnected but you could get false readings from a simple digital meter. Batteries often are the filters for crappy DC current and without a good filter (battery) attached, you may read high or low.
Trying to keep this very simple.
And if your battery cables are corroded, start by taking pictures and then carefully disconnect wires from the negative side and begin cleaning them up. Follow the negative wires to wherever they are connected to the frame and disconnect and clean that area too.