Is it permissible to advise a wife to divorce her ill-mannered husband who harms her and her children, knowing that she provides for him and the household, and this is out of fear for her life and the lives of her children, or is this considered incitement?

It is permissible for a woman to seek divorce (talaq) or annulment (khul') if strong justifications exist, such as her aversion to her husband's character or appearance, his deficiency in religious commitment, his withholding of financial support (nafaqa), or his failure to grant her conjugal rights, as in the hadith of the wife of Thabit ibn Qays. Since your sister is most knowledgeable about her husband's situation, she should assess the benefit of remaining with him for the sake of the children's welfare, as the presence of a father in the family is generally better than separation, and is worth patience and sacrifice. She can seek assistance through supplication (du'a) and by enlisting righteous people from his family to help improve his condition. If he does not improve and she fears for herself, her religion, or her children, then there is no blame upon her for seeking separation.