I'm having a fairly similar sort of issue with my series three. Whenever I hit a bump driving, some part of the circuit for the cold start light grounds, so it lights up. It appears to have absolutely nothing to do with cold or starting.
However, when I was replacing the dash, and instruments, I didn't see any switch back there behind the firewall... and I spent a fair bit of time re-connecting absolutely everything. I could be wrong tho, I didn't tear into the harness proper.
Could you maybe describe this switch? Or where I could find it?