I bought a selection of little switches to replace the original bits of plastic dust and such I found in the housing.
There's not much room.
No SPDT switch rated for the headlight current I could find would fit. No decent quality switch of any type I could find would fit. I got an Asian made el-cheapo radio Shack or equivalent SPST switch with open spring metal parts in back, quite small and rated OK for the solenoid current. It fit in the housing with no mods, the threaded part fit nicely in the button hole with just enough thread sticking out for the nut to fit. I connected the headlight to bypass the old shutoff-while-starting function and have seen no difference for its loss. This has worked for many years. Recently I learned that the button can pull out. After finding the button and asking my son not to pull it out again, it continues to work fine.