Also, if it seems like I have a sense of urgency on the issue, that's because he starts kindergarten (not sure if you call it the same thing over there, basically first year in public school) coming up in August.
Kindergarten here is preschool, he would be going into primary school. I would suggest that Judo would be really good for him, it's tough and exhausting with little standing around, it's very physical and good fun. It does require concentration of course but I think honestly it will help. This is one of UK's best Judoka and his story. Ashley McKenzie: how a childhood fight over a Pokemon card led to Commonwealth Games gold