Your school might have access to one in the district. If you can afford it, a private one would likely ensure more confidentiality.
If it were me, I would not involve the school. The school psychologist is working for the school district, trying to keep down costs. This would only stigmatize your son, I fear and result in little to no services or clarity.
If your son is referred to a neuropsychologist by his pediatrician t should be covered by your insurance. If it were me, I would either go to a neuropsychologist at a teaching hospital, in my state which is CA, that would be UC San Francisco or UCLA or the like or I would go to the child development center at a regional children's hospital. (Both of these options should take Medicaid and Medicare as payment, and have sliding scales.) That would be a start. But of course, there are private neuropsychologists who would be competent, if you could identify them.
We don't know your child and what he is dealing with. A good child therapist could help.
My son punched holes in the doors and hit the refrigerator. This behavior began after he turned 18 and I had no way to enforce bringing him to an evaluation and he did not cooperate.
In your situation, I might think about helping him channel his aggression and learn how to control it. Martial arts are one way. In Brazilian martial arts, for example, there is a Mestre who teaches a code of behavior and living, how to moderate one's impulses and regulate them and rise above them. They are usually good people. Two of these are Capoeira and Brazilian Jiu-jitsu. I would guess the Asian martial arts are the same. My son adored the Brazilian martial arts.
Or any other activity your son would like, where he is guided by a caring and responsible adult who is not a family member. One activity that is great involves horses and equestrian therapy. There is a nationwide group that provides this therapy. Horses are the very best therapists themselves. Because they are herd animals they are exquisitely sensitive and can transmit loving calm. I would do that in a heartbeat.
Art therapy is another possibility. Music, another.
But then, eventually, our children have full freedom of choice and live their own lives. You are very wise to act now, as you can.