IC: If the net result of expelling the child for lack of payment is that he/she will end up in public school and then possibly become irreligious, that should be allowed to happen if the school bill cannot be paid?
Is this out of town where there are no other choices of yeshiva/bais yaakov or can the child go to a different local yeshiva/bais yaakov?
Is this a modern or chareidi school? The modern schools are super expensive (possibly because of their expensive academics) and are known to be less willing to accept less than sticker price whereas the chareidi schools tend to be much more negotiable and a much lower sticker price to start off with.
Sam2, Succah 45b or Sanhedrin 97b.