Of course an amicable arrangement is always preferred.
But on this one, there isn't really a compromise. Either DD goes to local school or the one she's currently at 45 mins away. She can't go 50-50, she can't go to one 20 mins away.
The only possible compromise is if the root cause for ex's actions is not the school but something else.
Besides which, he (and his solicitor) are bluffing, the words are chosen carefully (assuming what you wrote is what they said). "they would try to take me to court on Tuesday to consider the matter.". They will "try" for Tuesday to "consider" the matter. His solicitor will be well aware family court issues takes months, sometimes years to resolve. By which time, DD is firmly established at her new school and no court would think its better. He could propose to send her to Eton and it wouldn't matter. So the threat is toothless and when someone's bluffing, it pays to call their bluff.
Plus if ex is working full time, chances are he won't be eligable for legal aid so every letter, phone call will cost him. Does he really have the stomach to fight on a paper thin case?
If DD is with you practically full time, he would be hard pressed to get an order stopping you from leaving the country, let alone defining what school she attends.
If I were you, I'd stand firm whilst trying to get to the root cause. Why is he so against the move? Is it control issues or something else? Is he fearing he will get less contact? If so, can you assure him another way?