Oddly, unplugging the LCM didn't cure the fault today, which is the first time that's been the case.

The only difference I could think of, is that I now have the BlueMusic bluetooth adapter fitted in-place of my CD autochanger. So I unplugged that, unplugged the LCM and plugged it back in, and the fault has gone.

Bluetooth Music Adapter BMW E46 E39 E38 E53 E85 for 3+6 CD Changer Connector 4251072613315 | eBay

So having the bluetooth adapter connected stopped the K-bus from resetting when the LCM was unplugged. One thing I didn't think of at the time, was to check whether my phone was connected via bluetooth and somehow keeping a connection alive?

My battery level is now low, so looks like I'll be taking the beast out for a drive during lunch to charge it back up