My son has been attending a small private school attached to our church. About a year ago, we requested of his teacher/principal that he receive a Dyslexia screening. The teacher agreed to get that set up. Months later it still hadn't happened. It eventually came out that the teacher had been deliberately stalling getting the testing.
I know that this would be illegal in a public school setting. I also know that the IDEA Act requires the public schools to work with private schools to some extent. What is less clear from what I can find online is whether (and under what circumstances) the private school is required to work with the public school and the parents to facilitate this testing.
Note: In this case, testing was eventually accomplished and the private school did facilitate the testing once we asked the public school directly.