whenever i try to launch the game, i get a black screen without any response from the game except the cursor turning into the custom stellaris cursor. I've tried looking up solutions online, but those are only for the steam version

Linux? How I fixed on Steam Deck with Bazzite OS:
Directly download the Linux version from GOG and not through a game launcher like Heroic or Lutris.
Step 2 ???
Step 3: profit
Extra steps If you don't know what to do with an .sh file:
If on an immutable OS (like Steam OS or Bazzite) I would recommend doing this through a DistroBox container,. Use your CLI from the folder the game is in to run: sh <file name>. Follow command prompts.
Now you can add to steam deck library by right clicking on the app icon