I would want to argue that he is definitely a flat character. Let us remember the central distinction between these two terms. A round character is one that is fully developed so that we are able to appreciate the psychological complexity of this character. A flat character is a character that has just one or two central characteristics and is therefore never really developed.
Harrison seems to have one characteristic and to perform one central action in the story. He is incredibly intelligent, able and handsome. His one act of rebellion is to challenge the basis of this dystopian society, that is based on achieving perfect equality, by showing his brilliance through declaring himself Emperor and taking an Empress, then dancing so that the world is forced to look at and admire their skill and beauty. Note what he says in his act of defiance:
"Even as I stand here--" he bellowed, "crippled, hobbled, sickened--I am a greater ruler than any man who ever lived! Now watch me become what I can become!"
He deliberately draws attention to his own greatness and what he is capable of being. However, that is it. We know nothing more than that. He is a character that seems to be peculiarly limited for one who is so brilliant. Therefore, because he lacks psychological complexity, we can say that he is a flat character.