Thank you for your suggestions. I have not yet mounted the microswitches, so I can't yet verify that they won't get damaged with any play in the bushings. I did order switches with extra-long levers, and the metal is flexible, but I won't know for sure until I tried. This is on my to-do list.
Optical detectors would be an interesting solution, though I haven't worked with them before. I'll have to look into that. I already wrote the sketch for using 4 switches, so I'm sure it would be relatively easy to integrate them into this project.
Would it be plausible to use a hall-effect sensor? I found inexpensive switches on Sparkfun that might work. While there's still clearance to be concerned about, they may be more flexible and definitely less archaic than my original plan.