I would pay for the sensory integration therapy with the OT, no matter what the cost. It has been a life saver for us. We went through it once for 3 or 4 months when DS was in preschool and are doing it again now 2 years later. The effects ARE long lasting...DS has been using the "strategies" he learned years ago and now needs an upgrade to more age appropriate strategies, so we are taking him again to outside OT. Invariably, when the school makes a negative comment about DS, it feels like a knife in my chest; but when I take him to see one of his outside support team (psychologist, OT, SLP, social skills group, ASD specialist, or psychiatrist depending on the school's complaint) s/he says the school doesn't know what they are talking about. We do a little fine-tuning and DS is back to doing ok in school. DS is 2e, unlike your daughter, so he has an IEP and support at school, but what school provides and what we can get outside are lightyears apart. We thought once he was in public school, our costs for therapies would be nothing, but the schools just cannot provide what private therapists can.
Unfortunately, although your school "seems supportive" they don't have to do anything. Even with our legally binding IEP in a public school, we have had countless meetings to "remind" the school that they MUST follow the plan!! I can't imagine trying to convince the school of what they need to do without DSs IEP...they would laugh us out of town. Since you are in a private school, can you have an IEP? If not, can you write out a "contract" with the school to get them to follow the plan you need for your daughter?
Nan