Since the Fly Proxy takes care of auto stop and start, you would need to configure services. An option might be to use Flycast to take advantage of the Proxy but on your private network only.
There’s some info on this beginning here in the original auto start and stop thread: Automatically starting/stopping Apps v2 instances - #29 by amine
Another bit explaining how it works with private networking here: Single VM vs. process groups vs. separate apps: current best practice for multiple processes? - #3 by ifiwasabird
Flycast docs: Private Networking · Fly Docs