Should the husband fulfill his wife's wish for divorce and khul' without her family's knowledge and with her remaining in America, or would her remaining divorced there lead to a greater corruption, especially with her fear of ill-treatment from her family upon her return to her home country?

Do not rush to respond to your wife's request for a Khul'. Be patient with her and give her enough time for her emotional state to stabilize. Remind her of the fear of Allah and the importance of the children and the good relationship between you. Try to understand the reason for her desire to separate, as there might be a shortcoming on your part. If separation is inevitable, let it be in consultation with her family, and ensure her return to them, while being keen on supplication and seeking forgiveness.