I would just set up the server for "serial console" and then run ordinary serial over existing(?) cat5.
All of the CAT5 runs from that cupboard are already being used for Gig-E, and only one of them goes to the study. Also, the PC doesn't have any serial ports (and USB to serial ain't gonna work until it's part way through the boot process, which is kinda the problem).
Hey ho. I was merely wondering. If I could find a cheap VGA-frame-capture device, it might be a fun Raspberry Pi project (in my copious free time).