Network “sign-in” webpage?

It would replicate this one specific group/chat only - from the moment of subscription. Depending on the load it can get heavy on your instance.

Matrix has one caveat: it synchronizes every room (group chat) from another instance to your instance fully- which one user subscribed to on your instance. Because of this the instance-systems/servers are under heavy load for private userage (not controllable number of users and chats). Many governmental institutions (controllable number of users and chats) use though, because in case of “disasters/incidents” the data is not lost but saved all over all replicas.

Use a dedicated DNS-service on your local network which has the local IPs in it’s DB. Use that DNS-service as your first/primary DNS on your local network (settings).