How can a father, (50% Legal Custody), fight for the right to have his child go to a better school (Private/Public)? Advise welcomed!

The bad thing with 50,50 agreement, someone is really 51 since someone gets to decide if the two disagree.
So in your agreement, if you don't agree with the other person, who has the right to decide, often in agreement one parent may have right in education, another medical and so on.

Next who has the physical cusotdy, do you share it also
i.e. who wakes the kid up, puts them on the bus, or drives them to school