All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Providing for young children is an obligation on their solvent father, and no one else is obliged to provide for them. Ibn Qudaamah wrote, “If the child has a father who is able to provide for him, then it is not obligatory on anyone other than the father to provide for the child.” [Al-Mughni 9/258]
Hence, you alone are obliged to provide for your children as long as you are able to spend on them. Your wife is not obliged to provide for the children even if she is rich and has a source of income. Please refer to fatwa 163965. As long as she is not obliged to provide for them, you have no right to oblige her to spend on them by the power of the state laws.
Also, the fact that she received a big sum for settlement at the time of divorce does not permit you to oblige her to do what she is not obliged to do. You willingly paid her the settlement money. The evidence that negates this issue is that she is not obliged to spend on the children according to the Shariah. So it is prohibited for you to take any of her money without her consent. The Prophet, sallallaahu ‘alayhi wa sallam, said, “The wealth of a Muslim is not lawful (for anyone else) unless he gave it willingly.” [Al-Bayhaqi]
Allah knows best.