To the OP: I've read the thread, and I am just another family lawyer, so you will decide whether to take note of what I have to say, or not.
Someone upthread said you cannot play by the rules when the other person is not - and to an extent, in child arrangement matters, I am inclined to agree. Particularly where the opposing party is an alleged abuser and appearing to use DARVO techniques. In the light of that agreement, and in that context:
You should take your little girl to school on the days you have her; you have justifiable reasons as to why this is in her best interests. I suggest you write to the father and set out your reasons for doing so, and copy his solicitor into the note/email (send it the day before school starts), and state you have issued a further urgent application to court to support the decision, as the delay is detrimental to the child. You may also wish to include a paragraph stating that you are willing to abide by any decision the court makes in the future, if that results in the school having to be changed, but in the interim, your child should start with her peers.
You should issue another application to court on an urgent basis for a decision on the school prior to sending the above note/email to the father. Ask for a hearing within 48 hours (make the application now - go to court with the form if you have to). Your solicitor should have knowledge of your local court and the judges, and be able to make a reasonable guess at the success of such an application (if I made a similar application at my local court, I could virtually guarantee I'd at least get a hearing - done it before on many, many occasions).
Whilst other lawyers on here have absolutely correctly identified that your child is not of legal school age, that does not mean that she does not have the right to start school alongside her academic year. A decision needs to be made one way or the other, and the court should not be delaying this, as delay is harmful to the child. The fact that the father changed his mind after the two of you agreed is a factor the court will look at, as well as the needs of the child in relation to the school type and which best suits her.
Yes, all the cautious responses may have a point, the father may try to retain her, or send her to the other school, but that could happen anyway. The court ought to have dealt with this element expediently and they failed to do so and left you all in this untenable position.
So you have your options: wait for the current application to be heard, or make a further application, and in the meantime send the child to school with her peers.