the user received a key, but for a duplicate of the game that he already has.

The winner did not say they already own the DLC, they were telling you that the key you provided is a duplicate - meaning it had already been used by someone else other than them. In that situation you would need to provide them with a new, working key.

That's why I suggest that he note that he accepted the key from me.

Even if this had been a situation where the winner already owned the DLC, you would still be telling them to break our rules. Winners are not allowed to mark a giveaway as "Received" if they already owned what the giveaway was for. If the winner already owned the DLC, told you to reroll and then you decided not to reroll, then the winner would need to mark the giveaway as "Not Received" to abide by our rules. But again, this specific situation is not about the winner already owning the DLC but that the key you provided doesn't work.

Or the second option is to delete the distribution and the key.

As stated by FateOfOne, winner does not have to agree to deletion. The default option when winner doesn't get their win is to mark the giveaway as "Not Received." Deletion is something a giveaway creator can attempt to do and winner can agree as a courtesy, if they want to but winner is not in any way obligated to agree to deletion.