Then it has to be subsidized somehow which puts another checkmark into the federal assistance category. There has to be something that says what they collect money for, an enrollment form, terms of service, terms of dismissal. When we're talking kids of mandatory age somehow it is on file with the state. They can't have no records at all. I'm sure you signed some sort of enrollment form, waivers, health care forms, etc. A contract is any agreement you signed on to.
You may be thinking of posh schools when it could really mean only grooming for a homogenous student population free of disability they would have to accommodate, a population that does not challenge the environment the administration prefers to cultivate rather than serve as a public accommodation. Whether the family pays or the family is on tuition assistance or the school receives monies elsewhere for that tuition, it's still not a personal favor.
The problem here is there are rules to 504, education law, disability law. You have to match the details of the situation to applicable laws to find which protections are available. While you are outside of the public system you will not be able to invoke the greatest of those protections.
It's going to be a tough set of decisions here, and I've walked what I'm talking because I just came from a small, private religious school of about 50 students max and not an iota of that decision making or negotiation was easy. It was definitely more tuition than $750/year hence why it was independent. I don't know how many would agree or disagree but I have no solution that would make them change in the way you want them to. You could invoke laws as long as you meet requirements, you can seek protections, but none of that will bring them to heel immediately not to mention happily.
Even a 504 will not mean you'll never have a violation or not need to tend to it on a regular basis or find situations that were not well enough covered in the accommodations that need to be addressed. In a small school with this much magnified social drama between parents and administrators it may not be feasible to call violations and remain in good graces with the staff or other parents.
The consensus here will be, yes, absolutely get a 504. But if you want FERPA protection, Title II and 504 with responsiveness from public district administrators who may be responsible anyhow for enforcing the 504 at a private religious school, the more long term solution is a move more directly under their auspices.