This might sound odd for those that don't know about App Images, but these self-contained programs are very powerful apps on Linux, but they don't play too well on Steam Deck outside of the Desktop mode.
Thankfully, there is a very simple solution, that will allow you to convert any AppImage into a runnable application you add to Steam and therefore run in Game Mode as well as Desktop Mode on Steam Deck, thanks to DevOpsIsAMindset on Reddit for originally pointing this out.
If you don't download it to the Home Directory, you will need to make note of the path for the next section.
Next, you want to open Konsole and enter the following
chmod +x appimagelauncher-lite-2.2.0-travis995-0f91801-x86_64.AppImage
If you didn't put the AppImage in the Home location, you will need to add the location after ./ above.
Note: if for any reason, the above command says "path not found" or similar, open KWrite and paste the commands in as shown above, then copy the filename of the AppImage you downloaded and paste in place of the name in the copied command, sometimes the format across devices and copy something that we cannot see, but will stop the command running.
Once the command has run successfully, you should have a new folder in your Home directory called "Applications".
Now, any AppImage you drop into that folder will automatically be converted and available to add as an application in Steam.
Using the Alderon Games Launcher for Path Of Titans as an example, in the Steam Library in desktop mode, open the Steam App, Library and select Add A game -> Add a Non-Steam Game and add the AppImage you put in the Applications Folder.
Switch to Game mode and you will now be able to run the AppImage without any further changes.