You can pick anyone you want to do an evaluation, but the district may have a list of "criteria" for instance not paying for a person who is charging exorbitant fees, is outside your geographic area, uses non-standard tests, etc. The district gave me a list of psychologists but I ended up going with someone off the list. Since it was a neuropsych/educational psych at a major university, the district couldn't really argue with it. I know that you just had an eval, but I think you need other tests that were never done. She was focusing on ASD and didn't do any language testing, EF testing, etc.
What you need to do is let the school district do an eval, and if you don't think it addresses your concerns, you write the sped director a letter stating that you don't agree with the eval, or it didn't address your concerns, and you are requesting an IEE at public expense, and to please give you a list of criteria for selecting an evaluator. The school district only has two options. 1) They take you to a due process hearing to prove their eval was appropriate, which can end up costing them more than an IEE or 2) They grant the IEE and you can take him wherever you choose (within reason). School district almost always do #2, but you have to let them try to do a decent eval first.
