My first response would be MANUAL. And this is why, when you are in AUTO, the camera is constantly adjusting for the lighting conditions, meaning if you go in the shade, if a person with a white shirt leaves the scene and a grey shirt enters, or if the sun moves from in front or behind, your camera will adjust. Now try to color correct in post. It will be vary hard because it is a moving target. If you color balance manual, you have a fixed setting that you can then make adjustments to as necessary. If there is a drastic change, take the time to re-balance (such as moving inside or stadium lights get turned on). That is my 2 cents, take it for what it is worth.

So what do you think about me setting the custom white balance once with my color checker's white card and then, about every hour or two, showing it in the frame so I can correct the video in post every hour or so (the event has long pauses in between performances) with a white balance correction based on that card? Would that be a good method?

I think you should run comparative tests. I think it depends on your camera's intuitiveness and "reasonableness" with auto WB, and whether that produces a pleasing result over the day's normal lighting shifts.

"long pauses in between performances"?

Based on your description, I think that's the best you can do in the long pause, showing WB card for reference.... Alternatively, I would play with Kelvin Temperature White Balance, if it is in between 5700K-3200K, I may take 4000-4500K, just going for closest white or possible in between that can be corrected later.

Of course, this is just my personal trial-n-error... but one thing for sure, I'll avoiding Auto WB as well.