If an overflow screen would be connected as an external monitor to your broadcasting computer (i.e. via HDMI), within OBS Studio you can right click the
Program window and select
Full Screen Projector to send that to the external monitor.
But HDMI, VGA etc. are not designed for long distance connections. The way to go to get the output to another room would be to send it over an IP connection using NDI. You'd require the
obs-ndi - NewTek NDI™ integration into OBS Studio plugin. I have no experience with that, but many do... so feel free to experiment and ask follow-up questions at this place.
