Since you don’t need to photograph, then @worriedguy is on the right track, but I don’t think you need to buy a stand alone prism. I don’t know if this would work, but it would be worth an interesting experiment, and I see no reason why it wouldn’t work. It runs in line with @worriedguy‘s suggestion.
Instead of buying a stand alone prism, just purchase an old film SLR camera from eBay. Any one will do, and they can be had for $20. all day long. Get one with a standard 50mm lens.
Mount that camera in the camera obscura pinhole, facing the subject. If the subject is extremely well lit, and the projection screen is in a very very dark room the light path from the SLR may very well be enough to project the light out the back of the SLR eye piece and onto your screen. Obviously, the higher quality SLR camera with a fast f1.4 or f1.2 lens would provide greater light transfer.
The lens/prism combo on the old SLR would act to reverse and invert the image properly on the projection screen. However, if the participants are behind the screen, the image would be reversed again. Have them stand in front of the screen. This might help if you use a highly reflective surface like a projection screen.