Is it permissible for the husband's brother to stay overnight in the wife's home alone with his mother and wife, especially since the husband is traveling and has forbidden him from doing so, and given the resulting embarrassment for the wife?

If the overnight stay of the husband's brother in the house leads to khalwah (unlawful seclusion) or fitnah (temptation/strife), then it is not permissible to allow him to stay overnight, regardless of whether the husband consents or not.

However, if the overnight stay does not cause khalwah or fitnah, and the husband consents, then there is no objection to it.

If the husband does not consent to his staying overnight or entering, it is not permissible to allow him; based on the Prophet, peace and blessings be upon him, who said: "It is not lawful for a woman to fast while her husband is present except with his permission, and she should not allow anyone into his house except with his permission," and his saying: "And you have a right over them that they should not allow anyone whom you dislike to sit on your furnishings."