" The specialist unit had written a letter to her GP, which said they were “only commissioned to provide a service to the mainstream schools”.
When A (person or organisation) "commissions" B (person or organisation) to do something, it can just mean A has asked if B would do it. However, often there is some money or gift given to B as part of the "commission". If I "commission" a painter to paint my son's portrait, that definately implies that I have paid the painter something or will pay the painter something for the portrait.
In this scenario, I would strongly suspect that in some way the clinic had been FUNDED to supply mainstream schools. It is common for county education departments to pay for a service to run for their children, or for individual schools or academy trusts to buy in to a service to let their children have access to it.
When the unit said they were "only commissioned to provide a service to the mainstream schools" it literally meant the money the government provided to mainstream schools was how the clinic was funded and therefore it wasn't fair for the clinic to serve a child not in a mainstream school, because they would be using money allocated for the kids attending mainstream schools locally to attend to pay for that child. In the decision to send your child to private school, what other services that school will provide and how your child will access them needs to be a consideration.