I went on to Zumo forums to try and remember what this Zumo 350/390 problems was with my RT. It is all to do with the Zumo keeping the canbus live and prevent it from shutting down. This then rapidly drains your battery.
Here is a copy of my posts from 2013:
I have just returned home after an expensive wasted afternoon at my local BMW dealer due to my battery running flat. ?150 later the cause has been identified as my Garmin 350 LM cradle staying live after i have turned off the ignition. My bike is a R1200RT which uses the BMW canbus system. Unusually this problem only happens when the satnav is not left in the cradle when i turn the ignition on. If the satnav is left in the cradle when i turn the ignition off i get a message advising external power has been lost and the device will switch itself off. Perfect this is what it should do. The canbus system runs for a minute or so after the ignition is turned off and then shuts down the power. The problem is if the satnav is not in its cradle, which is about 99% of the time. Today we discovered that after turning the ignition off the satnav cradle continues to draw about 5.4 volts even after the canbus has supposedly shut down. There does not seem any logic to this! It has drained my battery regularly in a day or so. Anyone any ideas what is going on. Its too late tonight to contact Garmin customer services but i will be on the phone to them first thing in the morning. Rich
I'm still waiting on my new power cable from Garmin and in the meantime a local bike mechanic had a play around with an electrical meter. What he found was if the bike was turned off with no sat nav in the cradle then for some strange reason the canbus system is being kept live. He got a steady 12v from each of the auxiliary sockets that just didn't turn off regardless how long you wait for the canbus system to switch off. He then unplugged the satnav cradle from the bikes wiring and hey presto the canbus system works normally again with no power at the auxiliary sockets. Therefore the garmin power cable is not allowing the canbus to turn off, unless of course the ignition s turned off when the sat nav is left in he cradle. It then powers down as normal and the canbus switches off. I will post the update after the new power lead is fitted to see if the same thing continues or not.