Am I obliged not to pray supererogatory prayers (nawāfil) during the period of 5 to 10 minutes before the specified time for Fajr, because this period was when the Messenger of Allah, may Allah's prayers and peace be upon him, used to rest in preparation for Fajr prayer after his night vigil?

There is no harm in performing supererogatory prayers throughout the night until the true dawn appears. If the dawn has indeed broken, then limit yourself to only the two rak'ahs of Fajr. There is no evidence for the obligation to stop performing supererogatory prayers during that period. What is established is that the Prophet, peace and blessings be upon him, would lie down after the two rak'ahs of Fajr and before the obligatory prayer, and he might even sleep during this reclining.