Also in the compatdata directory, but with very long numbers. You kinda have to search which one is which.

For a non-steam game, a random generated AppID is used to identify the application and name the prefix.

The only mapping I found between a manually added .exe and prefix is in the shortcuts.vdf file (somewhere in ~/.steam/). You need to read the file with a hex editor. It's the 32bit unsigned int after the "appid" identifier.

So yeah... probably easier to follow Ehvis advice and search for the folder or list the compatdir contents by date and open the latest created one.

Just for reference: Finding the prefix for a normal Steam game is easier since it's just the AppID for the game.

If you don't know it you can simply grep for the name of the game in the .acf files (one level above the compatdir directory). The match will have the Steam AppID in the filename.

Still, I would like to see an "Open Prefix in file browser" button in the Steam client!

The only mapping I found between a manually added .exe and prefix is in the shortcuts.vdf file (somewhere in ~/.steam/). You need to read the file with a hex editor. It's the 32bit unsigned int after the "appid" identifier.

What's weird about shortcuts.vdf is that the appids match those used for the grid images, but not for the directory names of the screenshot directories in .steam/steam/userdata/USERID/760/remote/APPID/screenshots

If you'd like to try it, I made a bash script a few years ago. Still works here. It's not perfect, uses Protontricks to read ~/.steam/steam/config/config.vdf. I offer zero support for the script, enjoy. :)
Non-Steam games will always create prefix in your home dir by default no matter where you install the game itself. And yes, Steam is notoriously terrible at deleting these dirs so you'll definitely have to do some manual cleanup.
Example output:
 
$ ./compatcheck.sh

Non-Steam apps - Note that these compat dirs may no longer exist.
Default location: /home/panther/.steam/steam/steamapps/compatdata/
                   -------------------------- | ---------- | ------------------
                   Protontricks detected game | Compat dir | Compatibility Tool
                   -------------------------- | ---------- | ------------------
                                              | 2387369984 | boxtron
                                              | 2372834471 | proton_7
                                              | 2538838069 | proton_7
                                              | 2728246381 | proton_experimental
                            DRG Survivor.exe  | 2384107741 | proton_8
                          GravityCircuit.exe  | 3834198777 | proton_8

And yes, Steam is notoriously terrible at deleting these dirs so you'll definitely have to do some manual cleanup.

I think that's a thing of the past since 2 or 3 weeks ago. Now it seems to delete the prefix every time.