There isn't any way to check other people's DLC, which is why SteamGifts will still show you giveaways for DLC you own. Kind of a bummer, and a weird choice from Valve IMO.

Speaking of which: I have the base game, but none of the DLC. Do you mind if I enter?

Oh, I see the issue now. I tried finding a solution but couldn't and now I read Octorocket's post on top of that. So my comment was... I don't want to say needless... but... helpless(?).

Only thing that gets tangentially close would be to check if someone won it already via SteamGifts tools, to at least exclude that the winner won it twice.

Thinking of it, maybe adding someone as friend on Steam and then checking the "x friends own this game" part on the DLC store page works? But then the winner needs to accept the request.

Aaanyways, have a nice day or night, whatever time you have while reading this.