• On your Steam Deck, go to desktop mode. Click the Steam button, go down to Power, and then Switch to Desktop. Wait for your Deck to reboot.
• Next, open up the browser of your choice, go to the Blizzard download page (or just search for it on Google), and download the Windows installer.
• Next, we need to add the downloaded .exe file to Steam. Open up desktop Steam, go to Library, and at the bottom left corner, click on the plus sign, then Add a Non-Steam Game.... Navigate to your downloads folder by clicking on Browse, and then go to home/deck/Downloads and add the Battle.net installer file to Steam. Don't forget to show All Files under File type: if you can't find the installer.
• Now, navigate to the newly added entry in your Steam Library. It should be called something along the lines of Battle.net-Setup.exe. Don't launch it yet, but instead click on the cog wheel icon, then Properties... and under Compatibility, toggle the Force the use of a specific Steam Play compatibility tool option. During the beta, it was important to choose GE-Proton 7.51 or Proton Experimental. Follow our guide here to learn how to install GE-Proton. But it seems, it now works without pre-selecting a compatibility layer.
• Now hit Play to launch the installer, follow the different steps, and wait until it's done. When the installation process is done, you can remove the Battle.net-Setup.exe from Steam again. Right-click it, go to Manage, then Remove non-Steam game from your library.
• Next, we want to add the installed launcher to Steam. In Desktop Steam,  go to Library, and at the bottom left corner, click on the plus sign, then Add a Non-Steam Game.... Now we need to look for the file. Click on Browse and navigate to the following folder by going through the folder structure: home/deck/.local/share/Steam/steamapps/compatdata/.  Don't forget to show All Files under File type: if you can't find the installer.
• Here you need to search for the latest added folder under DATE MODIFIED. It should be a 10-character long number. Open it, and go to the following folder: pfx/drive_c/Program Files (x86)/Battle.net. Here select the Battle.net Launcher.exe file. Add this one to Steam. If you can't see it, be sure you have All Files selected under File type:.
• Don't launch it yet, but instead click on the cog wheel icon, then Properties... and under Compatibility, toggle the Force the use of a specific Steam Play compatibility tool option, like we did in step four.
• Phew! We're finally done. Launch Battle.net now. If you want to customize the artwork, I propose you check out the following guide, but don't forget to rename the non-Steam game we've added.