How does TimeMachine store Calibre's metadata and backup files?
To manually dig through the paths, start at
/Volumes/[TM drive]/Backups.backupdb/[drive name]/Latest.
'Latest' is an alias - I'm not sure how that will resolve on nix.

The following all depends on the alias resolving correctly, as the actual locations are not 'fixed' but each may be in a different 'dated' folder on the TM drive itself.

Assuming that resolves correctly, then the Calibre plist is at [relative refs to your current folder]
~/Library/Preferences/net.kovidgoyal.calibre.plist.

Calibre Library itself is straight in your user folder ~/Calibre Library.

Prefs are in ~/Library/Preferences/calibre.