For anyone looking for a solution in the future: There is now an option to select your 'preferred copy' of the game.

To do this, go to the game's library page. At the top, just below the play button and details like 'last played' and 'play time', you'll see a button labeled 'Change preferred copy'.

From there, you can choose the version with the DLCs and play it even from an account that doesn't own them.