My uncle was taught to left hand from right hand because he injured his right hand and lost a part of his thumb when he was 8. He started stuttering, but it went away when he returned to using his left hand. It would be great if your brother could socialize with people who stutter too, or are deaf, or for any other reason prefer writing over talking, his age, face-to-face. Or with friends who would be patient with him and wait for him to write everything down. You might ask him if he'd like to visit self-defence courses, because learning how to protect himself and his personal space will boost his ego and helps him with controlling his emotions. I've seen good results with kids who are shy, have ADHD, PTSD or stutter (who started doing judo, fencing, tai-chi) Although he's fed up with therapy, he should practice alone every day, at least to get used to his own stutter. I wish you guys the best luck!