Try swapping a LED in place of the neon lamp. I do this on Fenders all the time - click vanishes. Cheap. My theory is the neon lamp generates a very sharp waveform when the gas breaks down and starts to conduct, and that radiates into nearby circuitry. You may have to add a series resistor to the LED to approximate the action of the original vibrato. I also do this on old Ampegs with failed vibrato lamp/photocell units.
I sort of doubt you'll find anyone who was listening closely in particular to Magnatone 15 in 1966 and remembers. BUT I could be wrong about that