Now it's OK. But it would seem Steam trolled me. Says I already own the product despite not having it in library. I verified and it turned out that I have the content of the DLC while not having the DLC itself due to having the Beta Backer DLC (not listed in store, only in the library, hence I didn't notice it at first).
So far, when I tried to activate a code and there was a message telling I already own a given game/add-on, I gave it away and the giveaways were marked as received, so you can ask for a re-roll for somebody else to enjoy (I still see the DLC as not owned on Steam, so I'd take it as a proof it'll work for sure).

Sorry for all this, I'll verify things more thoroughly later on when it comes to DLCs.