Is it permissible to set a trap for the wife and her lover in order to gain custody of the children, especially since the wife forces the daughter to call her lover "daddy," which has affected her psychological state, while giving the wife all her rights?

You should have chosen a religious and righteous wife. However, since you have married this woman and have children with her, you must try to reform her with wisdom and exhortation. If she repents and becomes upright, then keep her to preserve your family and children. But if she persists in her corruption, then divorce her and seek a righteous wife. If her bad character and transgression are proven, her right to custody is forfeited. You are permitted to spy on her to document her corruption if that is a means of proving what she denies.