Constipation in someone that young needs to be screened for congenital diseases. Did the hospital where he was born check his condition? Did he ever require an enema?
Is the problem lack of peristalsis or Bristol Chart 1 type of stool? If it’s lack of movement, tinkering with the diet will not help. You’ll fist need to rule out nerve problems like Hirschsprung’s disease, which will impair his transit and execretion, before considering dietary intervention.