Since proton runs each 'windows' game in its own container, by default the 'real' app doesn't get access to the save files in the demo (I assume there is some way in the steam dev api to change this - because I've seen it work fine on other games).
Here's a work around. Run this to copy all the saves from the demo appid to the real game app id:
cp -a ~/.steam/root/steamapps/compatdata/2822250/pfx/drive_c/users/steamuser/AppData/Local/StrikeTheEarth/Saved/SaveGames/Slot* ~/.steam/root/steamapps/compatdata/2541530/pfx/drive_c/users/steamuser/AppData/Local/StrikeTheEarth/Saved/SaveGames