The process of checking a DLC involves
Step 1: Add the winner as a steam friend
Step 2: Go to the steam store and see if you can purchase the game as a gift for the winner who is now your friend. If steam denies it, steam will tell you if it is because the recipient already has the item in their library.
Step 3: Do NOT purchase the game on steam for them (unless, of course, this giveaway is a steam gift). Instead,
Step 4: Send the key, or if they already have the game in their inventory (Step 2), ask for a reroll and provide the screenshot

Alternatively, you could always ask the winner to take a screenshot of the DLC page--including their username, the url, and the place where steam would say "this item s in your steam library". Maybe also ask for them to include the clock to show the date that they took the screenshot.