I get the impression that this is being done client-side, so the problem may not be detecting or communicating with a private LAN server; it could be the clients themselves being able to communicate over the Internet with Microsoft's servers.

I would be surprised if the system went through the server at all. There's a separate online component for verifying accounts and downloading skins/capes, so that's what you'd need to circumvent. That was possible way back in the day, but I have no idea if you can still do that.