Thanks, but I’d prefer open-source rather than professional.
Have you conaidered a £10 HDMI capture card? Not open source, or much of a project, but it would Just Work.
I've got an Elgato Camlink, but need the freeze button (e.g. CamLink input to a Raspberry Pi full-screen, feeding through to HDMI output, with a GPIO button to pause the live stream).