No way around this that i have found. If you have the proper DNS settings that allow the FQDN on the local network that is about the best you can do.
I suppose you might modify the host files on all the local clients to direct to the local IP when the FQDN is used. not sure how this would interact with the launch center. Never tried it.
DNS is not the issue. I have updated DNS on our domain so connecting to the server using its DNS-name (=FQDN) is OK.
The problem is that the server does not use this name when it presents itself to the clients.
Like I said. Host file changes might work.
Launch center will only auto look for local hosts using IPv4 or IPv6 so far as I know.
I put in a feature request for adding a favorite host via assisted install but not the best answer.
Maybe if your DNS server is assigning unique Names on the same FQDN (machine1.domain.com) to all the clients it might work. Never tried. Just throwing out some ideas.
Can you add the host DNS name as a Favorite?
Edit: Never mind. I see you already tried that.