Is it permissible to divide the Dhuhr prayer such that the sunnah prayers before it are prayed first, then one returns to complete the fard prayer and the sunnah prayers after it later?

If the teacher does not permit leaving to perform the prayer, then you must perform the obligatory prayer within the available time, along with the Sunnah of Dhuhr if time permits. However, if leaving is permitted, there is no harm in performing the pre-Dhuhr voluntary prayer (ratibah) half an hour before the obligatory prayer. It is best to perform the prayer at the beginning of its time, except when awaiting the congregation or during intense heat.