What grade level?
To deliver special education services is very costly for a school system and there are major incentives to try to duck it.
I can assure you that local to me there are kids in private schools K -12 that are recieving special education services but it can be a very daunting task for the parents and requires addressing a complex set of due process rules. (And not to mention a few parents who seek to get elite private schools at public expense by twisting some rules)
I also suspect the local school may assume since you can afford a private school you will soon tire of the process and pay for some private solutions.
I have no clue if your childs needs are such that an IEP is appropriate to address what she needs so as to get an appropriate education.
Personally I never addressed how to fight for childs needs from outside the bubble --I did spend a good amount of time in due process hearings as to students inside the bubble. I know the steps are there but suspect they won't help you find them (though the law says they must show you the steps)
One starting point is to go to your local public school administrative offices and get a copy of the due process rights as to special need students --all the steps are supposed to be in there --trust me they may leave out a few or make it difficult to follow
By law the public process is required to address the special needs of students necessary as to a FAPE no matter what it costs the them --and a $1 million per child per year point was passed decades ago --I'm just not up to speed any more as to how this all fits private sector schools .
IF there are serious needs to address : I think you need to NOT rule out that you enroll student in local public school so as to make maximum use of the rules --or at least have a straight poker face --you could enroll student in local public school and 10 minutes later file to paperwork that sets in motion evaluations headed for a due process hearing.
Locally --unassisted parents get a lot of lip service and some perfunctory appropriate services often worded in a way that protects the service provider against failure. Odds favor those who know/learn the ropes and better yet use a professional guide or gladiator --IE an attorney with a track record of prevailing as to due process/special education cases from student standpoint.