[2022] Bought game on Steam, Ubisoft Game Launcher asks for Key

Dear community and/or support,

I bought Assassins Creed Valhalla in May directly via the Ubisoft Games Store (without any DLCs).

Now I also wanted to buy the DLCs for Christmas, so I bought the Complete Edition on the Steam Winter Sale (just buying the DLCs doesn't work on Steam).

So far so good.
I deinstalled the game via Ubisoft Game Launcher, then Installed the Complete Edition via Steam.

When I first tried to run the game via Steam, I ran into the issue that Steam didn't open the Ubisoft Game Launcher, which i could solve by granting the game administrator rights.

Now when I run the game via, it opens the Ubisoft Games Launcher, which then asks for a game key (which I of course don't have) and I can just cancel the window, which results in the Game Launcher not recognizing my game as installed, and I'm not sure if the DLCs are added to the Game Launcher.

I tried the following:
1.) deleted the cache folder of the Ubisoft Game Launcher - no effect.
2.) reinstalled the Ubisoft Game Launcher application - no effect.
3.) tried to localize the (Steam) game folder via Ubisoft Game Launcher - doesn't work, just shows a error message.

Anything else i can try?
I would be very grateful for tipps.

Merry Christmas everyone! (bit late, but I think it still counts)