Although those physical approaches are fascinating, I think there must be an easier software solution. How hard can it be to live edit the webcam recording so that half of the frame is a static picture? Then you just have notes flat on your desk... And the webcam will show it empty.

Alternatively, one could run a VM with the monitoring software, and have a separate window with notes. As long as the mirror is angled so to hide a small portion of the screen that should be good enough.

To build on your physical solution using polarization: that's easier. You can apply a polarized filter to the webcam (you can get one from an old pair of 3d glasses) and then have the notes on a secondary screen that will then look completely dark on webcam. Maybe lay the device on a black background so it's not obvious that there's a "turned off" device and you're done.

This is all a fun thought experiment. I don't have invasive professors, thankfully. They simply made all tests open book and raised the difficulty of the questions accordingly. That being said, I enjoy trying to work arounf hypothetical limitations...