Have you tried speaking with the pediatrician for a referral? Perhaps they would know of a psychologist that would happily perform the testing outside of the school system. We are going through that now for additional assessments and the pediatrician was able to refer us out. I’d also suggest maybe reaching out to other parents of special needs kids to see what services/organizations they used in your community. A local support network can really be helpful in helping navigate the system.
And I agree with the other posters who have suggested Wrightslaw. That was a terrific resource that I used continually when we were having issues with the IEP.