I’m afraid I’m out of my depth if we are not talking about the original hardware.
Obviously too late now, but I would have wanted to get the old switch to work correctly rather than introduce another element in the system.
I recently had a similar problem with the 6 starter switch. It transpired the Tufnol cam was badly worn and, in the end it was a simple fix.
The 4-cyl switches I have had were simpler with just a sprung leaf contact. They tended to suffer from pitting of the fixed contact combined with deposits accumulating. That was generally cured by polishing with fine emery. In extremely pitted cases, the fixed contact could be rotated to present a fresh contact surface.