Two different issues.
1) Dispute over school selection ... If the orders are silent, then this falls under joint parental responsibility (assuming both parents have JPR) ... You could potentially take it to a court having family law jurisdiction. Need to undertake dispute resolution first & be issued a s60I certificate....
Is the child already attending a school contrary to the wishes of one parent?
2) School fees .... If there is a CS assessment in place then both parents are already contributing towards school fees on the basis of that assessment ... If the fees are high, a parent can apply for a reassessment under
2.6.9 Reason 3 - high costs of caring for, educating or training the child in the manner expected by the parents | Child Support Guide
Courts don't get involved in child support & fees unless circumstances are such that a normal formula assessment can't be done by CSA.