I'm having very similar problems with my '02 540iT. This LCM/K-BUS sound like the culprit, but I have no idea where to find them in my vehicle.
Here's my story:
A couple weeks ago I was driving and noticed the turn signal indicator stayed on for a few seconds after the signals were off, and the radio and computer turned off at the same time. That was the tip of the iceberg...
The following have since only rarely and briefly worked correctly:
Turn signal indicator and clicking do not work, but outside signal lights still work;
When set to Automatic, there is a delay in turning the headlights on, then they don't turn off when I exit my garage;
Reverse lights do not work, nor does the always-on rear wiper function when reversing, (but the tilt-down of the right side-view mirror works perfectly);
Trip functions of computer (is it still called an OBC?) work in the driver console, using the turn-signal stalk control, but Check Control function and center console display (along with radio) remain dark and do not work;
Many of the warning systems (probably all controlled by the Check Control part of the computer) such as the coolant level, selv-level "inact", key in ignition, headlights on, don't function;
When the systems do come on, it is often just the lights of the radio/computer, but no sound or functionality.
A possible cause in my case may be that about a week before all this started almost a full gallon of water spilled in the back. I didn't have time to do more than sop up what was in the carpet, so over the next couple days I ran the heater on high with the vents pointed to the back and the windows open. When I had the time to clean it up there was a puddle in the spare tire well - very close to the air compressor and related wiring for the Electronic Height Control - which has been only slightly more glitchy than prior to the spill.
Any helpful hints would be appreciated!