Leave the battery disconnect turned on.
I had a problem this past year of a failing battery. The symptoms were that the 12 volt lights would dim slowly until the converter would turn on. When it kicked on you could hear the cooling fan run and the lights would get bright for a few minutes. All of this while connected to shore power. This tells me the 12 volt lights, furnace and other items do actually run off the battery but the battery is charged through the shore power.
I would turn off as much of the 12 volt draw as you can to see if the shore power can get the batteries up to full charge.