Is it permissible for a wife to ask her husband to watch her dance to songs and music under the pretext of him refraining from looking at what is forbidden, knowing that he refuses to listen to singing because it affects his heart?

You are not permitted to obey your wife in listening to forbidden songs and music, because "there is no obedience to a creature in disobedience to the Creator." You must prevent her from doing so because you are responsible for her and her guardian, as Allah Almighty says: "O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones," and as the Prophet, peace and blessings be upon him, said: "Each of you is a shepherd and each of you is responsible for his flock." The fact that she is a Sharia graduate does not mean infallibility or legitimize her actions. Rather, she should be more committed, fearful of Allah, obedient to Him, and an advocate for the Sharia. Furthermore, her specialization does not negate your guardianship over her.