Who is the flat character in Catching Fire?
It depends how the question is intended.
If it asking which character in Lord of the Rings is physically
flat, then Sauron comes close since he is just a flaming
eyeball-ring.
If it is asking which character in the Lord of the Rings is not
terribly well-developed, most of the undeveloped characters are the
auxiliaries. In the Lord of the Rings, we don't know much about
Elrond other than him being old and sagely, nor are we well
informed about Halbarad, Aragorn's second-in-command among the
Rangers. Eomer also remains flat for most of the story, being
little more than a slighted prince until he ascends to kingship and
mourns his father, cares for his sister, and weighs the future of
his people.