I don’t see a resolution without battling it out with lawyers.
Defining “had to place child in private school” in a way that is not a personal preference of the person placing is going to be a real longshot.
Generally speaking, if it was legally necessary due to a disability, the IEP team would compel a public institution to fund it. If that is not the case, it would be considered personal preference in my experience.
If you browse this forum, there are a lot of conversations about the benefits of private (customer service, rigor, etc.) versus public (more legal rights, cannot kick out if skills are low, etc.).