By coincidence someone else posted about the exact same problem as this on the same day, over at bimmerfest, and included what might be a hugely important piece of information.

His diagnostic showed an error in the instrument cluster reading "Latch-up measuring device driver". That rang a bell for me and might point to the source of the problem. A "latch-up" is when a parasitic circuit forms in a microchip, leaching electricity away from the pathways it's supposed to follow onto another pathway unrelated to its function. When this happens the chip will no longer function properly, but the problem can often be solved by power cycling the chip. And THAT sounds like exactly the symptoms I experience.

So now my new question - Does anyone know of someone who does repairs on instrument clusters in Montreal?