I think you need to detail your setup and what you're trying to stream to.
Original or New Livestream?
Camera, too vague, too many types, be specific
MacBookPro, be specific, too many types with different connections.
Windows, why, specific needs?
Capture output from the projector? Don't you mean capture output from ProPresenter on the MacBookPro?
Describe in enough detail so that someone could build a matching system with the exact same equipment. Then you can get a detailed response.