Yes that sounds like part of the same issue. If it is the trickle charger then it should stay good. If off the charger and not being used 1 week might be a rough average among the responses I hear.
If you need it to stay live then the big red switch on the front of the electrics/fuse box in the engine bay (in front of the steering wheel sort of) can be flipped to kill the car which tends to keep it all live. This will isolate everything so can make audio/trackers a little jumpy etc.
Inspect the wires around the door to body joints for fraying and exposed cable as they don't seem best protected.
On the Supersports the roof panels are known for leaking a bit and need some love and new rubber seals on an infrequent but regular schedule. If the water is coming in through these and down the pillar into the dash area then it won't help.
I would guess you may have a trapped wire or lose connector. Fastest way to resolution is probably a pure electrical specialist rather than most of the generic dealers.
