You might consider going through a children's hospital for the testing. That's what we did, and the hospital was able to find the right testing situation for us, plus knew what additional tests my son would need to figure out his quirks.
We had the same insurance catch-22 as you, in that if he didn't end up being diagnosed with a LD, we would be responsible for the entire bill. It was frustrating, but as we were fairly sure he had dyslexia, we took a chance and it ended up costing us about $200 total. Does your son's psychologist think he has a LD? That might help influence your decision one way or another.
I agree with your opinion on school testing. The school isn't required to do what's "best" for your son; just what's "fair" and "appropriate" (or is that adequate?). So I wouldn't count on them to unravel a complicated LD.
For two years, my son's school had told me he was at grade level on reading, or within the range of normal, when I knew he should have been a good reader by 2nd grade. But the fact is, grade level expectations are so low in the younger grades that there's no way we could use that to judge whether he had a reading disability, which the teachers were doing.
I think if you can swing it, you should go the private route, especially since there seems to be a countdown in your case.
