Yeah you can Google how to install wow on Steam deck and follow the guide, with a caveat that on the steps between installing battle.net and creating a launcher for it on Steam after it’s installed, I suggest moving the contents of the proton bottle to a shared space so you keep you credentials. Let me get on my pc in a few minutes and I’ll get you some instructions.
Add it to Steam from the Games > Add a non-steam game to my library…
Right click on it from Steam library, Properties…, Compatibility, check “Force the use of a specific Steam Play compatibility tool” and select Proton Experimental. Close the window.
Run the installer by double-clicking it in your library. Go through it as usual, make sure you uncheck to start it with Windows, and to mark Keep me logged in.
Install WoW (don’t need 100% installation, just start it), and click on the cog icon and Create a desktop shortcut (no shortcut will be created in your desktop)
Open Battle.net settings and in App, On Game Launch, set to Exit Battle.net completely.
You can also mark When clicking X, Exit Battle.net completely.
When done, close it fully (from tray and etc).
Navigate to ~/.local/share/Steam/steamapps/compatdata and find the folder with the Battle.net installation (it’s going to be the one with a longer name, and most recently modified).
(Optional, see footnote) Move the contents of the pfx folder somewhere else like ~/.local/games/proton_prefix/pfx and create a symlink from ~/.local/games/proton_prefix/pfx to ~/.local/share/Steam/steamapps/compatdata/XXXXXXXX/pfx:
Footnote: The reason for moving the proton prefix folder away is that this way you can have a shared proton prefix for all your non-steam proton games with the advantage of keeping a shared login state and etc between the apps since the registry is stored inside the pfx folder, but have a separate shortcut for each in your steam library by always creating this symlink back to the shared folder, and the ability to tune proton settings to each different application separately as those settings they are kept in the parent folder.