I found an interesting reply on Proton's github. Apparently there is a way to run non-steam games with just the Steam client.
Do these steps once:
Go to your Steam settings, "Steam Play" on the left and Enable Steam Play. Optionally, Enable it for all titles too, so that you need fewer steps for each game you want to add (though I suggest against it, see below).
After Steam restarts go to the "Tools" section of your Library and
install any Proton versions you are going to use. If you don't know,
select the most recent one (currently Proton 4.2-2 is the
For each game:
Go to Steam's main window, on the left-down corner click "Add a Non-Steam Game...", "Browse...", change the File type to "All Files", find your game's Windows executable and add it to your library.
(Optional) Right-click the new entry, go to its Properties, name it
as you please in the first field and select "Force the use of a
specific Steam Play compatibility tool".
Note that if you didn't choose to Enable Steam Play for all titles in the settings, this step if REQUIRED otherwise it's going to run using your system's wine version.
Even if you selected that, you may want to try different versions for
each game and see what works better. Remember to have the
corresponding Proton version installed from Tools, otherwise, it
falls back to system wine again.
If you look on the wine official website, you will find that Starcraft II is supported out of the box with Wine. This means that all you have to do (in theory) is install Wine, then install StarCraft II, and it should work without any playing around with the configuration.