Lutris Doesn't Recognize Cuphead's DLC - Steam Version

I recently discovered that many games on Steam are DRM-free, so I wanted to transfer them to my local Lutris library (I’m using the flatpak version with Ubuntu 20.04).
I was able to do so with various games simply by copying the folder with the game’s data that’s in the /Steam/steamapps/common folder in a new location and by configuring Lutris to run the executable (add game → add locally installed game). I was able to run both native and Windows games this way, I only had to add the steam_appid.txt file to some of them and to grant Lutris flatpak the access to the /.local/share folder because some games save the progress there by default (it’s not ideal but honestly I don’t know how to fix that).

As the title said, when I copied the Cuphead files and configured Lutris as described above I was able to play the base game, save progress etc, but it didn’t load the DLC even though in the folder with the game data there are the DLC’s files too.

Has anyone encountered a similar problem or know how to fix it?
I’m quite new to Lutris so any help is very appreciated, thank you!