A new tribute from an unnamed district could be the main character. This would give the author more freedom to develop the character's backstory related to claustrophobia. For instance, this new tribute grew up in a district where they were often forced into small, cramped spaces for work, which led to their claustrophobia. And now in the Hunger Games, they have to face it in a much more dangerous and life - threatening way.
Peeta might also be a good choice. He's a character with a lot of depth. If he were the main character in a claustrophobia - centered fanfiction, it could show a different side of him. Maybe he gets locked in a small room by the Capitol as part of their mind games, and his claustrophobia becomes a major part of his struggle, both against the Capitol and within himself.