Schools should at least be able to tell if child has a "disorder of written expression" even if they don't label it as a disability. If the school doesn't know anything about dysgraphia, I'm not sure they would be competent enough to choose tests that would pick up the issue.
"Should" is one thing. "Can" is another.
Our experience with school psychs is that they have Master's degrees, and they are about as qualified as a psychometrician. That is, they are not equipped to do differential diagnosis as a neuropsychologist would be. They also often do not have access to or expertise about the specific assessments that would best help you pinpoint issues.
If you really want to know what's going on, I'd use a private neuropsych. If you really want school to act on what's going on, I'd use a private neuropsych in parallel with the school's eval process, and provide the school copies of the private data for anything that you really want done correctly.
