Very good troubleshooting
EVERY time that light came on full brightness, the fuse would have blown. (you did not have to waste a bunch of fuses to get to this point)
You also determined that the engine does not have to be running for the fuse to blow.
However: You did not mention that the light comes on bright when you push the 'safety switch'. Perhaps play with that switch some more and try to isolate the problem. Perhaps the wires leading to it are touching each other or the frame?
Have you looked for wires 'chafing' on the frame as the treadle is rocked?
You could try hosing down that switch with WD40. That may help free it up a bit.
Perhaps you could remove the switch and with the wires still attached, work the switch by hand and see if the light comes on bright at any point.