Is it permissible for the wife's brother to divorce her, or must she remain under his responsibility, and he is not sinful for not cohabiting with her as long as she is content with that?

Conjugal relations are a right for both spouses. If they waive this right, there is no sin upon them. However, each of them has the right to retract their waiver and demand their right in the future. It is better for your brother not to divorce his wife as long as she agrees to forgo conjugal relations, based on the Almighty's saying: "And if a woman fears from her husband high-handedness or estrangement, there is no sin upon them if they make terms of peace between themselves; and peace is better." So, if the wife waives all or part of her rights concerning maintenance, clothing, or lodging, and the husband accepts that, there is no sin upon them, and reconciliation is better than separation.