Like it or not, the Watsons are in this together. They're a family first, and they tackle the world as unit. They love each other, hate each other, irritate each other, and try to understand each other. When the conflict in The Watsons Go to Birmingham—1963 finally reaches critical mass (i.e., Byron's trouble-making has gone too far), the Watsons turn to their extended family for help. While at times it might seem like the Watsons don't get along at all, underneath all the bickering and bullying and tattling, the love this family has for each other saves more than one of them in the end.
If Byron stays in Birmingham, it will just break up the family, and that wouldn't be good for the Watsons.
Taking Byron to Birmingham is the best thing for the family.