We are in a quandary as to what we should do and would appreciate any suggestion by you. The Christian school does not feel summer school or tutoring will get him up to speed with their second graders.
If your son switches to the private school and is required to repeat first grade, he is likely to be considerably older than his classmates. In fact, you might want to check out the ages of the other students who would be in the private school first grade with your son next year. Consider also that your son will be 19+ when he graduates from high school if he repeats first grade.
It definitely won't hurt for your son to have some tutoring this summer even if he has to repeat first grade. It could be that some of his basic skills are weak and need to be improved for him to do well at either school. You might even consider having your son tested just to pinpoint any problem areas that could be worked on this summer as well as to get a better picture of his skills. Plus, there is always the possibility that the private school's test scores don't reflect his true ability.
Once you have made the final decision, work with your son's teacher and the school to make sure that he keeps up with the other students in his class. Neither public nor private school is fun for a student who is struggling.