What’s the name and location of the file that I need to transfer which would allow me to save a relay’s phase progress in case of an OS reinstall?
Have you read the documentation? Topic 9
https://community.torproject.org/relay/setup/post-install/
You need the config that should be in /etc/tor/torrc and the keys from /var/lib/tor/keys
Please have a look at I want to upgrade/move my relay. How do I keep the same identity? | Tor Project | Support
And maybe How exactly do I backup my relay's identity keys? also helps you.