What is the ruling on preventing one's brother from divorcing his wife and searching for another wife for him, given our knowledge of his insistence and the wife's desire to stay?

You do not have the right to prevent your brother from divorcing his wife. However, if your intention is to urge him to keep her and not divorce her merely out of dislike, then there is no sin upon you; rather, you will be rewarded for that. You should advise him to fear Allah and fulfill his wife's rights. There is no objection to her waiving some of her rights so that he does not divorce her. There is also no harm in helping him marry another woman and choosing a righteous one for him, while enjoining him to be just between them.