Jason,
The inverter based charger will only work when the inverter is connected to a 110V source, i.e. shore power or generator. When driving, w/o the generator running, the batteries are being charged by the engine alternator. As far as leaving it on while plugged in, your charger should automatically switch from bulk to absorb to float charging status as the batteries become full charged. At the float level, the charger is only supplying 5A or less so it doesn't represent a very big load. The inverter is probably pulling more than that w/o the charger. If you don't need the inverter running when in storage, you can rely on the solar charger (trickle charger) to keep the batteries minimally charged. However, after a period of storage, I'd recommend using the boost switch to start the generator and let it run about 5 minutes before starting the engine (again use the boost switch) and keep the generator running for at least 30 minutes to reduce the load on the alternator.
Steve