Is justice required between a wife from whom you have separated, but who is still in your home due to her betrayal, and the second wife, and what is the Islamic legal position on this relationship?

If the wife errs, disobeys her Lord, and neglects her husband's rights, then it is obligatory to advise her to sincere repentance. If she repents and becomes upright, then keep her and treat her well. But if she persists, then separation from such a person is desirable, and divorcing her is permissible. As for the marital relationship, it remains until separation is proven, and a mere feeling of divorce does not remove the marriage bond. As long as the marriage bond remains, you must fulfill her rights of lodging and intimacy, and it is not permissible to abandon her in a way that harms her. So, it is either holding together with kindness or parting with excellence.