The professionals who recently completed the evaluation should provide you with a clear interpretation of the results of the testing. In my opinion, the testing is not complete until you (and your son) understand the findings and how they relate to school performance. It is particularly important to know if your son has a learning disability and if so, the nature of it. This is true whether you decide to stay in the public school or have your son go to a private school.
While the private school may be the very best option for your son at this point, it's important to make sure that the public school has done all they are required (by law) to do for your son. You say that they want to retain him. This may mean that they don't consider him to be a student with special learning needs, since they are probably aware of the research that shows that retention is seldom a solution for such children (or any children, for that matter).
Has he ever been referred for special education services? Did the school do a comprehensive evaluation before you had it done privately? Were you informed about your rights under the special education laws in your state? If these things didn't happen, you should contact the special education department in your son's school and request these services. If your son has been identified as a child with special needs, and you feel that the services have not adequately addressed his needs, then you might want to contact the Florida Department of Special Education. Ask them how to find an advocate to help you make sure that the school is doing what it's supposed to do. If the school hasn't been keeping up their part of the deal, then you may be justified in having them pay the tuition at the private school. Let us know what happens.