Is it permissible for a brother to take revenge on his abusive brother-in-law, who withholds financial support, beats her, and insults her, or should he let his sister return to him and endure the humiliation and beating for the sake of her children, and will he bear her sin if he insists on her not returning?

If your sister's husband beats her, humiliates her, and insults her and her family without justification, then he is an oppressor. It is permissible to convince the father and the sister to seek a divorce, as long as the oppression is prevalent. However, you do not have the right to discipline him yourself. Your sister has the right to bring the matter before a judge, so that the judge may deter the husband or compel him to divorce. Not every divorce is evil; rather, it may be good for both spouses.