This is mainly a projector-technology issue, not just a camera issue. DLP projectors often don’t display the full image continuously; some show colors sequentially. A camera shooting live video can sample that cycle at the wrong time, which causes banding, color breakup, or flicker.

For still photos, a longer exposure can average the projector’s refresh/color cycle and reduce artifacts. For live video, that’s much harder because the camera frame rate and shutter are often close enough to the projector timing to reveal the problem.

If you need clean live video of both the subject and the projected image, the practical solution is usually to use a different projector technology—typically LCD rather than DLP. An LCD-based projector is less likely to produce these artifacts on camera.

Before buying anything, test your intended camera/projector combination if possible, since results can vary by model and settings.