Hi guys.
Thanks for the inspiring thoughts. I ended up doing something…alternative.
Seeing as this project is terribly limited (I have to choose 1 browser to work with, it has to be under Windows, and all information, connections and code must be available directly through the browser, with little to no load-time on any Internet connection), I ended up creating and embedding a small Java applet, which simply holds a ServerSocket running, and whenever we send it a message via LAN, it requests the remote server for the queue-number, and sends it to the JavaScript, which can then use it to draw the new number. Using a bit of concurrency trickery on the server, I should never be without the right number on the screen for more than a few ms.
I realize there are ways to get around the spiderweb of “origin policy”, but this seemed the most unintrusive way to me. At least I’m not going to violate any security protocols in any browsers anymore, and applets are already fully allowed on the client-PCs.