How can one deal with a husband who wishes to bring his homosexual brother into the home, and what is the legitimate Islamic ruling on this action, especially with young children present in the house?

If it is confirmed that the husband's brother is deviant and engages in the acts of the people of Lot, then it is impermissible for the husband to allow him into his home or to permit him to meet his children. It is also impermissible for the husband to ask his wife to show him respect. Rather, it is obligatory to shun him and display aversion towards him. The Companions unanimously agreed on executing anyone who commits the acts of the people of Lot. We agree with the questioner's stance on preventing him from entering the home, and the husband must make the same decision to protect his family. If the brother repents sincerely and his repentance proves genuine, then there is no harm in allowing him into the home, provided the wife commits to not appearing before him or being in seclusion with him. For the brother-in-law poses a greater danger, and he must be dealt with extreme caution even after his repentance.