Is testifying in court to show the truth and help the second wife obtain her rights considered a mistake, given that this led to the severance of kinship ties by the brother who divorced his wife because of this testimony, and swore never to speak to the witness or his sister?

If the situation is as described, then by your testimony, you have both borne witness to the truth against your brother. It was not permissible for you to refrain from performing the testimony related to the woman's rights if no one else would do so. By this testimony, you are not severing kinship ties; rather, your brother is the one severing them. You are obliged to maintain ties with him kindly. If he insists on severing ties, then the sin is upon him.