I think Brinker remains static. He doesn't change much throughout the story. He wants to see Gene confess his crime about Finny in the Butt Room in chapter 7, and then pushes it again in chapter 10 & 11.
He's a smart, tough guy the whole time who wants to enlist when the time is right. That fits a tough guy. He is also a moral character, he wants truth to emerge.
He is not a flat character, as mentioned above he has many qualities or traits. Brinker takes on the whole crew at the Winter Carnival when they try to steal the hard cider. This too matches his tough guy mentality.