The answer is Yes, you could do that; i.e. remove/disconnect the converter, and use a battery charger to recharge the coach battery.
You will need to make sure you do not over charge (boil the battery) or under charge (run the battery low ) These will both kill your battery in a short time.
The converter not only keep the battery charged, but it actually supplies the 12 volts to your systems when your plugged into shore power or running on a generator.
Without the converter, your battery supplies power to the furnace, water heater, frig, lights, fans, etc.
I don't like large power fluctuation getting into electronic equipment, connecting disconnecting a battery charger will give these fluctuation. Also hydrogen gas will be developed when you deep cycle the battery, so use caution when connecting/disconnecting the battery charger (
Keep an eye on the fluid level in the battery (you should be doing this anyway, but more so if your deep cycling the battery.
My recommendation would be to replace the converter with a three or four stage unit. This solves all the problems.
I've delt with these folks, they stand behind their products: