Private schools are a little trickier to get a SD into than public schools. One thing a lot of people fail to understand is that taking a SD to school is not a title III access issue. It is not the same as taking a dog to the grocery store. The burden of proof on the PWD and the standard for "reasonable accommodation" are higher than they are for a public accommodation.
I think the confusion comes largely from the language in the codes of federal regulation that reference some parts of public schools as public accommodations. Areas where the public are welcome, such as sports events, are public access under title III. However, areas to which the public are not generally welcome, such as individual classrooms when class is in session, are not under title III, but title II (for public schools).
Access standards are higher for "long term" interactions and relationships with the service animal team. A visit to a grocery store is a relatively short interaction. Renting an apartment or getting a job involve much longer interactions, usually years long, and many hours each day. School is more like a job or housing than it is like a public accommodation.
The school must take into consideration not just the needs of the PWD student, but all of the students and staff. In a short interaction, allergies and fear cannot justify denying access of a SD team. HOWEVER, in a long interaction, in some instances, it can.
If there is a child at school who is allergic to dogs, we must consider how the presence of the dog will affect that child's education. Allergy medications can make a person drowsy. If it is not possible to relieve the child's symptoms without impairing the child's ability to learn, then the accommodation isn't really reasonable. Ideally, the school would work to find a solution that considers the needs of both children. Perhaps they could be put in different class rooms and schedules instituted that would keep them from being together.
The whole point I'm trying to make is that the key here is REASONABLENESS. Access with a SD at a school is not an absolute right. Actually it isn't at a grocery store either, but the assumption is that because it is so minimally intrusive that generally a grocery store should just permit service animals as standard practice.
This is a hot topic where emotions can run very high. If everyone involved keeps focused on reasonableness and accepts that compromise will be necessary the odds of resolving school issues will much higher. The problems I've seen arise when the parents put their foot down demanding rights they don't actually have, and refusing to dialog at all (ie the Cave case). If the family of the PWD is exceedingly reasonable and makes their communication in writing instead of over the phone or to the press, they stand a much better chance of getting what they want if it does go to mediation or court. This is one instance when taking the high road can be critical to success.