In the abstract, it is a fine idea.
Concretely speaking, it does not fit in this story, because the doors are so perfectly unidentifiable. If the doors had unique names or numbers on them, I could imagine there is a puzzle to be solved, such as you suggest. But The Giant would have dropped hints well before this point in the story.
Call the Chekow's Anti-Gun. When no mention of a gun has occurred in the story, you can't have a hidden gun pulled out from behind some books on the shelf in the third act, and used to immediately kill an important character.