I your situation, with shore power, I would put a Battery Tender on the starting battery, rather than disconnecting it. This device has more general use than a Trik-L-Start which is just for the RV.
Battery Tender Jr might, or might not work. This lower rated model is sized for motorcycle and lawn mower batteries, may or may not work for larger sizes depending on loads and battery conditions. I can put the Junior I use on my motorcycle on my Honda Fit after it has sat a couple weeks, and it will come up to full charge in 2-3 days. But same device didn't work for my brother's Focus SVT, because the "phantom" loads didn't leave much for charging and maintaining the battery.
Situation with your charging battery suggests it is sick. It might be made to be useful much longer with a maintenance charger like Battery Tender or BatteryMINDer, but not with a cheaper trickle charger. Difference is the device monitoring charge state and charging only when needed. An always on trickle charge can dry out a battery.
