If you were doing this, consider cribbing from https://github.com/obsproject/obs...

If you were doing this, consider cribbing from https://github.com/obsproject/obs-studio/tree/master/plugins... which offers a variety of solutions including some rather exciting looking process injection (called "game" there).

I wonder if "entire chat app functions as OBS plugin" would work? Would solve the AV streaming side of the functionality.