Have you done these things already? When I had gunk in my shifter it made shifting intermittent, didn't do anything with battery. I presume you've tried a different cable to the problem shifter?
Havent just yet, I'll get the spray today. I tried different cables and all that...
Another thing I read elsewhere (but didnt resolve my issue) is that if you dont disconnect from the phone app, the bluetooth module will continue to drain the battery.
Have you done these things already? When I had gunk in my shifter it made shifting intermittent, didn't do anything with battery. I presume you've tried a different cable to the problem shifter?