Is it permissible to divorce a wife who repeatedly commits acts forbidden by her husband, such as displaying herself and wearing tight clothing in front of men, despite her repeated promises not to do so, knowing that the husband feels extreme aversion towards her and is unable to trust her, and that her family is trying to reconcile them?

You erred when you were lenient in choosing a religious wife, for a righteous woman is the best provision of this world and she safeguards her husband in his absence. While divorce is permissible when needed, help your psychologically ill wife seek treatment, be patient with her, and advise her gently. If she does not improve, then divorcing her is better. It is not necessary for the divorce to be finalized through a sheikh unless you are ignorant of its rulings.