Sorry to hear about your dilemma,
Strictly speaking, your daughter telling/asking you do something in contravention of the orders is not a proper basis for breaching the orders, and the court may take a dim view of you influencing your daughter to breach the order.
That said, the orders don't appear to prevent your daughter from asking her mother, but ultimately it is up to the mother to agree to any variation in the orders (though even she cannot do so if that would contravene a IVO).
If you will not talk to the mother on the phone and she is pressing you to call her before she will agree, I can only suggest you write to her asking nicely and setting out why it is in the best interests of your child that she be able to attend, and asking her to confirm with the school that you can pick her up. If she fails to comply then this may prove useful evidence if you have to again apply to the family court for a variation on the current custody arrangements, since it will make it clear that she is not prioritising the interests of your daughter. It may be worthwhile mentioning this to her if you think such a threat would assist in obtaining her cooperation.
If she refuses then legally you are stuck. Whilst you can take legal action to get new orders, and could in theory apply for urgent orders, in practice it would be very doubtful you could get suitable court orders in place in time to fix this problem, though knew orders addressing functions like this could be obtained for future occasions.
Whilst you could simply breach the orders if you can convince the school to let you take her, you would technically be in breach of the orders, and whilst I suspect if the matter were raised with the court, the court would be sympathetic, there are no guarantees.
I trust the above assists your understanding.
Good luck and please rate my answer.