If the solar panel has a solar charge controller, then the solar panel should be disconnected from the controller *before* the battery is disconnected from the controller. On reconnecting, install the battery and connect the controller first *then* connect the panel to the controller.
The panel may be so small it does not have a controller anyway.
Perhaps that solar panel is meant to maintain the engine battery, which usually is not getting charged when the house battery is from the converter when you are on shore power.
You could use your battery tender on shore power to maintain the engine battery and leave the house battery on converter on shore power while at home, and leave the solar to do whatever it does.
If you leave the old battery in till Spring, keep an eye on things in case the battery has a fault develop that makes the converter run hard and develop problems of its own (not likely but can happen)
1. 1991 Oakland 28DB Class C
on Ford E350-460-7.5 Gas EFI
Photo in Profile
2. 1991 Bighorn 9.5ft Truck Camper on 2003 Chev 2500HD 6.0 Gas
See Profile for Electronic set-ups for 1. and 2.