Can you share an example shot?

Fascinating thread this.

Is there any way to reduce spill from the projector onto the stage? I doubt a filter would be helpful.

A polarizer filter might reduce reflection from the stage floor. It won't help the color temperature issue. Replacing the 3200K bulbs with Cool White or Daylight will. Share a picture?

I would recommend starting at the projector level rather than the camera. Consider a less reflective screen for the projection screen. Pretty much any thin fabric will be able to be seen from the other side, assuming the projector is sufficiently bright, but there are some fabrics that work better than others. If rear projection is required, that limits you a little bit, but there's still ways to get creative and work around it. You want a screen that reflects as little as possible... you can experiment with different shades of grey fabric until you find what reflects the least light while still retaining good image quality.

If ambient light is still a problem, then perhaps your projector is turned up too bright, you have more projector than you need for the room. Consider dimming it and, when it dies, downgrading to a lower lumen projector. Just because it can get that bright doesn't always mean it should.

If you really want to get fancy, check out holo-gauze, you can see straight through it, but it makes a screen appear seemingly in mid-air. That wouldn't reflect down on the stage at all, but it also comes at the cost of the projections also passing through the screen and projecting onto the room around it, so projector placement is key to making sure you pull off a really cool effect while not blinding people.

I specialize in video projection mapping buildings and stages for events, as well as transparent projection effects using screens like holo-gauze or water screens, so if you have any projection questions, I'd be glad to help.

Here's an example of white projector light being shone through a polarizer. In this example at a certain angle you do seem to see warm white. The white balance of the camera seems off. But does it keep this polarization when bouncing off your subject, which is what you need for your test to be a possible success.

Related, but not directly answering -- I purchased 4 Tiffen circular polarizing filters during COVID for my four cameras, to see if they'd help reduce glare on facemasks worn by stage performers. I never even tried them since in tests on car window reflections I saw little to no success in removing the reflections?! In contrast, I used them in the past when filming a national Synchronized Swimming event and similar filters did an outstanding job removing water reflections.

@todd-b that sample video is definitely interesting -- implies it could indeed help!

Here's are two questions

1. Why are you white balancing your cameras for the projection @5600K and not for the people on stage @3200K? We always, without exception, balance for skin tone since that's the thing that will look most unreal to viewers; if a projected graphic or text doesn't look the right color usually it isn't as noteworthy. (obviously, there are exceptions when the color of the projected image is really quite important, and is unavoidably directly behind the talent)
2. Have you checked to see if your project has the ability to change its white balance? Many higher-end projectors do have adjustments for that.

Dealing with amateur stage lighting is a tough problem no matter what you do. Get that allot in community theater where extra bulbs are bought on sale and the actors change color as they walk around stage. You can always split your clips by zone and try and give each zone a custom white balance in post. But that's allot of work for every Sunday.

If all the lights are the same color temp, just white balance with an actor holding a white or grey card below their face near-perpendicular to the floor but angled downward just enough for the card not to send shiny spots into the camera lens. If they settle on a color temp closer to daylight, like 4000, you won't have to re-calibrate as much for shows that start with light coming trough windows and continuing into the night. And if the lights are color-changing, ask the lighting guy to humor you by setting them all to white. And let the colors fall where they may during the show. Make sure the follow spot is on if they use one. Also make sure your cameras are not set to auto anything except maybe focus if it's quick. But if your camera has an auto-white-balance that locks on record, that's another exception if it actually works.

Thanks to everyone for your helpful comments. I since read the polarizing filter works best with light at 90 degree angle, and the screen bounce is maybe 45 degrees. And I would probably get grain with the relatively low light.
 

I asked for screens on the side, or rear projector or LCD. Of course there’s no money for that. So it is what it is.
 

What you read and what experienced stage producers will see are different things. Since you are unable to share stagelit photos as was twice suggested, I agree; It is what it is.

I'm not sure how much this will help, but the picture on the left is ungraded, and the one on the right is the best I could do to color correct. The white line at the top is the bottom of the screen that is reflecting down on the stage, as the projector is mounted above on the light bar. Notice the backlight on her hair from the screen bounce. What I wish I could show you is the same frame with the projector off for comparison, but that didn't happen.

You could say the camera's color balance was off, but it's spot on when on the podium. And AWB was locked.

I've also asked that they raise the screen, or even use 16x9, to get that white bar at the top out of frame, but that wouldn't be according to the 3rds division of the monument wall.

Well, if you have any ideas, that would be great. But there is only so much one can do to work around poor lighting.

I'm not sure how much this will help,...

I am. That's why I asked for them. Your eyes and monitor may interpret them warmer, cooler, or about the same as this suggested starting point.

You could say the camera's color balance was off, but it's spot on when on the podium.

No, one correction works fine for both; yes, there is a slight skin tone difference that your customers will not find distracting. And neither will you, after three weeks or so without looking at it.

A possible starting point is shown. If you are willing to make the investment of time and training to learn to correct color in post, it will be rewarding. Fifty years and still learning.

Hint: Try testing your camcorder White Balance on the stage-lit projection screen first. And then lock it. Shifting White Balance ("Auto") and stage lighting in any of its manifestations are not your friends. No, a polarizer will not help with your lighting.