I improved my editing workflow and mixed the sound from the appropriate camera. Instead of using the workflow described in the Blender manual for the multicam effect strip, which has you splitting the multicam strip, I used a driver to animate the active camera from a single multicam strip. I used the same variable to control drivers on the four audio tracks I recorded as well, so the audio was always synchronized to the video. Editing the active camera curve in the curve editor turned out to be a lot less fiddly than working with a million little strips.

This one is in 60 fps! The video was also downsampled by OBS instead of Blender and I think the end result is the same or better and the captured video is way more manageable.