The battery supplies power for the lights, water pump, etc. when not plugged into AC power. The converter charges the battery and supplies 12V power for those things as well (when plugged into 120V power or when running the generator, if equipped). An inverter performs the opposite function, turning 12V power from the battery into 120V power. You want to avoid having the inverter power the converter ever as that is simply a colossal waste of battery power.
This 12V system (the "house" or "coach" system) is separate from the Dodge 12V system (the "chassis" system). Often, however, there's some sort of a device that selectively connects them together so that the house battery gets charged by the engine alternator while the engine is running. Often this is a high-current continuous-duty solenoid (relay) that connects them together, with the control line tied to ignition power or something similar but a bit more complicated (a voltage detector, a delay timer, or whatever). It can also be a high-current diode which permits electric current to flow in only one direction.
If you want some literature on your particular converter,
here some is. This is a fairly typical ferroresonant converter, by appearances. It has a pretty high float voltage (assuming it's working as designed) which would tend to be hard on batteries and in particular cause water loss at a pretty high rate. It also requires a battery to be connected to clean its output up to a somewhat tolerable level.
I have no idea how Triad-Utrad stacked up with other converter makers; regardless, replacement with a modern multistage one would be a good upgrade. One good option, not necessarily the only one, would be a Progressive Dynamics PD9245.
