Presumably because the existing contract is between the Childs two parents (it's never been OP or the mums husband on the contract), so no other person would be liable to pay the current fees other than the two people named in the contract.
The mum has refused to pay, so the DP is paying her share for the time being to keep the child in school, the school aren't 'going after' anyone as they are being paid.
The problem now is the change in circumstances as the child now lives with them full time, and the school need to re-assess to see if they meet the household income criteria for the bursary (they won't, but OP is refusing to submit her earnings anyway), and they are also asking her to now be the other person on the contract accepting liability to pay, it would make sense for it to be OP, but they can't make her sign it and she shouldn't have to, the responsibility to pay should fall solely to the child's parent, but it's unclear what the schools contract actually says.
They can't go after random people in other houses where the child doesn't live for payment, his mother is also off the hook as he no longer lives there or has any contact with her.