Oh well that's embarassing, I've only gone and bloomin' fixed it. And without resorting to using the optical sensor!

Short version: I rewired the microswitch board to work off the NO contact (instead of NC), pulling a 3.3V line from the unused BLTouch port on the motherboard. That's completely cured it, with homing repeatability in the sub-5um range and no more occasional big errors.

 The schematic below shows roughly how the circuit looks now:

I had to cut a track and solder a resistor onto the microswitch board, and swap wires in the connector to provide +3.3V from the BLTouch port to make it work. So it was a little complicated, but cost nothing and took about 10 minutes.