Since your BIL is so desperate for Divorce, why not use it to your sister's advantage to get a better deal for her in terms of alimony and custody. Realistically speaking, there is no point in dragging a dead relationship forward. It will only cause more pain and agony to both people. Ask for custody of the children, their maintenance and a lumpsum alimony for your sister. If he refuses to part with that, then refuse for Divorce.
Either ways, it is important for your sister to have proof of his relationship. Just a statement that he is cheating is not going to suffice. You need to have proof of the same, through pictures, messages etc.
My advice would be for you to try to settle the issues amicably first. if that doesn't work and he sends the papers for Divorce, then oppose that tooth and nail. Produce proof of his Adultery. If you are able to substantiate your statements with proof, his Petition will be dismissed for sure for a person cannot take advantage of his own wrongdoings. Your sister can also file a Petition for maintenance.