Is it permissible for a girl to falsely and slanderously accuse her paternal aunt's husband of assault—with details too shameful to mention—and agree to swear on the Qur'an, to appease her father who forced her to do so?

Obedience to the father in what he requested is not permissible, and Allah will provide relief and a way out. If the daughter fears for herself, she has the right to refer her case to the judiciary or security authorities to protect her from her father's aggression.