Video Broadcast within Retirement Community
Short of doing a live stream which you've already ruled out, your best bet sounds like it will be a fiber link between the auditorium and the health center.  HD-SDI has a technical limit of around 100 meters, so you would need a booster in between to make use of SDI (which would be a coax run and possibly a bit cheaper).  Neither of these options can be completed for less than a few hundred dollars in cabling alone.

Other than that, your best bet is to provide training on how to use the live stream and possibly setup a device like a roku that can be adjusted remotely over the network by someone in the main auditorium.