RetroPie requires all games to render without a window manager, so I don't think that would work out of the box.
It's probably possible to build RetroArch with X11 support.
v4l2loopback kernel module
uv4l-raspidisp might also work in theory. Apparently the PlayZone PZ-HDMI can take the video and loop it back into the Pi for casting. Some of these options are rather complex or costly.
For the best experience you could also use an HDMI to USB capture stick and connect that to the PC, but if you've got a long cable running to the PC that's not casting. Or for the best experience you would run RetroArch on the PC itself
Hopefully others have some better ideas.
Edit: Realized the RealVNC's framebuffer capture would indeed work here. Note it's not enabled by default on Buster. Go to the VNC Server tray icon -> Options -> Troubleshooting -> Enable direct capture mode.
Edit 2: UV4L comes with an RTSP server, so uv4l-raspidisp + uv4l-server may give you everything you need.