Hi John,
I would suggest that you leave the battery connected. If you disconnect it, everything that runs on 12V will be powered directly by the converter (battery charger). The converter is not designed to be a stand alone 12V power source, it is designed to be a battery charger. All the 12V things like lights, water pump and the controllers for the fridge, water heater and furnace should have a strong steady 12V supply from the deep cycle battery. The converter will put back whatever power is taken from the battery, at the proper rate.
Rob