request testing for your son in writing, use email. Print out a copy for your records. Sometimes if you ask (verbally) it is like you did not say it.
Even with a paper trail, follow up on it. Ask when the testing will be done- ask to sign an evaluation form- they have 30 days from the date you sign to do testing.
Which tests are used may depend a lot on personal preference of the therapist, but you also have the right to request a certain test if you feel it is more appropriate or decline a test that you think is inappropriate. The school did both academic performance tests and a Gilliam Asperger's test. Later outside of school he had CARS and ADOS I think. That was when he was diagnosed with HFA or "mild autism" is the term she used.
I would do both the school testing and your own outside testing through your insurance.
Do you have a Regional Center for services for him? They will do their own testing to determine if he qualifies. (Make sure to ask that your son be observed in the classroom, with his peers, or with peers in a "playdate" type setting, like a Boy Scouts get together, or something like that..
My son had one observation with a psychologist in her office, and then a follow up with another psyschologist in the school setting. The second report really "saw" his issues, while the first report did not.
We just recently got (not many) hours of "respite" care per month. It can be used at a local center where they have recreational drop in program after school. They work on social skills, and every little bit counts so we'll take it
Also, use the term "Autism Spectrum Disorder" instead of "Asperger's" on his IEP. It is accurate and atleast in our school district this seems to carry more weight.
If your son were to get the diagnosis, it could certainly help the speech therapist address what areas of weakness she should be focusing in on during therapy sessions, i.e. voice modulation, maintaining a conversation, facial expressions etc.