Before you make the final decision on how to proceed, look thru the forums here and you'll see numerous comments/threads about owners leaving their units plugged in to use the on board charger and their batteries end up boiling dry, dead etc.
The price of a BatteryMinder is cheaper than a new battery IMO, and you have to do something for the chassis battery, it's just as easy to rig one up for the house battery as well. I've never lost a battery using the BatteryMinder, in the Class C we had or the 4 batteries I have in my boat. The cranking battery in the '08 Class C was the original battery when we bought it used in 14, and was weak. A BatteryMinder brought it back to life and was going strong when we traded it in early 2018. I did keep it plugged up when stored at the house.
'08 Greyhawk 31SS Traded in
TV 2018 F150 FX4 SC 3.5EB TP/MT