Hello!
About the launcher unable to update itself or always applying the same patch over to the game, the trick is to use the "STEAM_COMPAT_DATA_PATH" environment variable in the launch options of Steam for the launcher .exe file you added as a non-Steam game.
This environment variable is used to change the Proton "prefix" to use.
Steam created a prefix when you ran the launcher's installer using Proton, and installed the launcher in this prefix. If you then add the launcher the same way in Steam, it will create a new prefix for that "game" when you run it as well.
I will roughly describe the steps to install the game correctly on Linux, using Steam. Read it thoroughly before doing anything, as understanding these principles will help you in the long run using Linux

You will then be able to troubleshoot pretty easily.
* Run the launcher's installer using Steam, DO NOT RUN THE LAUNCHER DIRECTLY AFTER
* Find the launcher's location using the following command from a terminal
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find ~/.steam/steam/steamapps/compatdata/ -name "OctoLauncher.exe"
The result will look like
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/home/linuxuser/.steam/steam/steamapps/compatdata/2669118820/pfx/drive_c/users/steamuser/AppData/Local/Programs/OctoLauncher/OctoLauncher.exe
Note that the first part is your prefix:
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/home/linuxuser/.steam/steam/steamapps/compatdata/2669118820
* Use the full address to add OctoLauncher.exe to Steam, as a non-Steam game (do not run it yet)
* Change the launch options of the newly added game in Steam (right click > properties > launch options) to the following (of course, you will have to replace with the prefix you found earlier):
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STEAM_COMPAT_DATA_PATH="/home/linuxuser/.steam/steam/steamapps/compatdata/2669118820"
Don't forget to select some version of Proton in the "Compatibility" section of the game in Steam. You can now run the launcher from Steam and the issue should be resolved.
Of course, if you have already set up most of this and got the game running, but only want to fix the launcher issue, you can simply update the launch options to use the right prefix.
I hope this helps!