After spending a while looking for my saves so I could back them up (hackers are out there folks!) I decided to make a quick guide that will hopefully help other Linux players back up their own games.

The excellent PC Gaming Wiki has helpful information that can point you in the right direction as well.
Be sure to check the footnote about Steam data.
https://www.pcgamingwiki.com/wiki/Dark_Souls_Remastered#Save_game_data_location

Your location may differ from mine. My distro is Bazzite 41.

My Dark Souls save was located here:

/var/mnt/games/SteamLibrary/steamapps/compatdata/570940/pfx/drive_c/users/steamuser/Documents/NBGI/DARK SOULS REMASTERED/67744560/

I hope this helps someone else.