IMO, the OP meant the 120v output to connect his new converter/charger to. But then he said the F20 has 120 at input but nothing coming out so how can he get 120 for the converter from that end of the inverter pass through?
I seems like the F20 could be yanked and a converter but in its place using the 120v input and the old DC output wires from its charger section.
But then there is no inverter. How is 120 going to get to the rest of the rig?
Once you have 120 to the rest of the rig solved, you could get an inverter and also a converter and make up your own inverter/charger easily enough for less money than a new inverter/charger.