It would work. It just activates the DLC part. It's up to the giveaway creator to restrict it or not.

As Kegfarms says, you can register the key without issue, but if you already have the basegame, then only the DLC will be added. It is up to you whether you are happy with this (for example, the basegame might be expensive and the DLC cheap).

In this case, Layers of Fear basegame has been given away for free in the past so I suspect a lot of people will be entering just for the DLC (as am I).

If you are not happy then check your winner and request a reroll if they already have the basegame. Also change your description to make it clear whether people can enter just for the DLC or not.