Hello temmas2012.

If a DLC's information is not available in Steams API then there's no way to check for it.

There's other things to consider though: The sites rules require you to only enter giveaways for games that you don't own, yet. So you should have checked before entering. While I understand that hardly anyone does that and instead simply relies to the site's settings that are just meant to help with that (which has it's limitations as explained above), there are tools that make this a really easy task. For example the Steam Web Integration script.
It shows you a green checkmark besides every Steam Store link for titles that you already own and a red cross if you do not own a title, yet. That way one can directly see if something is already owned or not.
I can only recommend using SWI. Not only to make things easier for oneself in order to follow the rules but also to help avoid unnecessary tickets and thus trouble for yourself and the giveaway creator.

You should hide giveaways for the game so this doesn't happen again.