Yes you can, it is just a matter of cleanup after that point.

You will get a lot of duplicate frames, it is just a matter of a script to clean them out of your final capture.

If doing that, then set up a hall effect sensor to count how many frames there should be, so you have a reference for any that are missed.

You could in theory I guess trigger a beep sound of a few ms to use as well, it would go onto the audio track next to each 'real' frame, might make it easier (or might not work at all, I'm thinking out loud).

If you got the projector to run at 15fps or 10fps and you are shooting with an NTSC camera, then cleanup may be easier too.

You may have to go back at the end and recapture a handfull of frames, but it would be much faster with a video camera. There will be compression artefacts, but the quality will still be very high. Capturing at a slower projector speed like 10fps would let you average multiple frames together to reduce noise and other non-image data.

If you can borrow a DSLR, you can use a single high quality lens, and video capture the uncompressed HDMI output, no compression problems then, but you will need an SSD to keep up with capture.

Either way, make sure you capture a set of flat frames, i.e. blank frames of white. You can then use those frames to offset any vignetting or dust motes on the sensor etc. and get a near perfect image.