D..
I want to recommend that you ask your ped to refer you to a speech pathologist who works with stutterers. He needs an evaluation. Your health insurance should pay for this eval.
You might wonder why you would get a private evaluation. Your child will need an IEP that includes speech services. The school has to go by private eval recommendations. The school's speech therapist would probably actually appreciate having a private evaluation to go by, to be honest.
I am not going to sit here and tell you that all private speech therapists are better than school therapists. However, for more difficult cases, a private therapist who has experience can be make far more difference than just a speech therapist who mostly deals with articulation issues.
My son had both private and school therapy. I will tell you point blank that my son would not have normal speech today (he's 19) if he had only used a school therapist. He had some specialized issues that not every speech therapist encounters. And not every speech therapist encounters stuttering. I moved heaven and earth to get my son help, going through my health insurance, fighting with my insurance company at some points, etc. AND I supplemented with school speech therapy.
Maybe this isn't what you want to hear. Maybe you are hoping that you only have to work through the school. I don't believe that you will get all the help you need like this. Please, get a private evaluation. Your ped can help you find someone. Or, if there is a children's hospital in your area, call them up and ask them to recommend one of their speech therapists who deals with stutterers to do an eval.