The school offer a discretionary bursary.
He won't get it with your income.
They want two adults named so that if one defaults they have back up. It's a private organisation so they are free to make those rules.
You and the school are both being fair.
DH needs to recognise that if SS mother was so bad, he did the right thing in getting custody. However, he cannot new send him to this school in the current circumstances.
He needs to console himself that changing school is the lesser evil.
Your fight is with DH not the school. He needs to find a guarantor if he wants his son to carry on going to that school.