Hi momcat. Mom of a kid on the spectrum as well as an occupational therapist who spent 25 years in the school districts (so I've sat on both sides of the table!) here.
I've had similar issues with my guy. In regards to academics (I'm assuming you're in the states), having a child who scores well academically makes it tough for many schools to recommend support, as technically they are required to provide an "appropriate and free education" and on paper the child is learning at a level commensurate with their peers.
With that being said, we all know there is more to school than reading, writing and 'rithmatic! And that is the piece you should emphasize at your meeting. Participation, social interaction, ability to produce work, using the skills being taught, etc. Has his teacher shared any concerns about any of these areas with you? Be sure to bring them up at the meeting. Let the CSE know you understand that he while he currently tests well academically, deficits in these areas will impact his ability to continue to do that and that is unacceptable. While you cannot ask the district to do a specific test (the district gets to pick the tests they administer), you can ask them to screen him for behavioral and emotional function in school (for example, the BASC is used for that purpose in many schools, but there are others).
And to clarify some things, the school does not have to test a child if a doctor recommends it, as they are not a medical facility. Their charge is to the educational environment and that's where your concerns need to be targeted. If testing indicates support is needed, they are then obligated to provide that support, whether or not their caseloads are full.
Lastly, yes. Getting the right diagnosis can be horribly frustrating (I have been there!). I knew he was on the spectrum from early on, but he didn't get any diagnosis until he was 5, and that was ADHD (the neuro at the time said he didn't quite make the spectrum diagnosis and that he was "quirky". Not helpful). He wasn't confirmed as on the spectrum to me until this week (although apparently his previous neuro DID give him the diagnosis 2 years ago; just never passed that info on to me).
But don't let that stop you! You don't need a diagnosis to give your kid the tools he/she needs to help them. For example, I've always used "countdowns" to give my kid an opportunity to transition (he does NOT handle change well LOL). So when he was young, if we needed to go do something, I'd let him know "We need to do X in 5 minutes", then remind him on 4 minutes, etc. He's older now, so I don't need to count down each minute, but I still point out the clock and remind him in X minutes we will need to do X (although usually I will have to point out time is up). When you have a free moment or 2, do some research, and definitely ask here, on what tools you can use to help your kid.
HTH!
Alli