Is refusing to care for the children of one of our troublesome kin who stay overnight with us considered a sin, and will the grandmother's prayer against us be answered if we refuse, knowing that we are compelled to do so?

It is permissible for you to refrain from caring for the children of relatives and to refuse their staying overnight with you, and there is no sin upon you in that. The grandmother's supplication against you because of this refusal will not be answered, for the supplication of parents is answered if it is due to disobedience or injustice. However, it is better to be patient and seek reward from Allah, as there is kinship and benevolence in it.