Is it permissible to repeat the 'Ishā' prayer as an Imam for one's wife with the intention of making it up (qaḍā'), or is there another intention for that, or must one stop leading her in prayer in this situation?

What is meant by qada' (making up a prayer) is performing the prayer after its prescribed time has elapsed. It is not permissible for someone who has already performed an obligatory prayer to repeat it with the intention of it being an obligatory prayer, without a valid legal reason, because the Prophet, peace and blessings be upon him, said: "A prayer is not to be repeated twice in one day."

However, if one repeats it with the intention of it being a supererogatory (nafl) prayer, there is no harm in that. Indeed, Mu'adh ibn Jabal used to pray 'Isha with the Prophet, peace and blessings be upon him, and then return to his people and lead them in the 'Isha prayer. And the Prophet, peace and blessings be upon him, said to the two men who had prayed in their dwellings and then came to the mosque, ordering them to repeat the prayer: "Indeed, it will be a supererogatory prayer for you."

Therefore, there is no harm for the questioner to pray 'Isha with his wife while intending it as a supererogatory prayer, not as making up the obligatory prayer.