Is it recommended to pray the Sunnah prayer of Fajr close to the Iqamah (call to commence prayer) due to the greater reward in doing so, or is that an innovation?

What your companion instructed you to do is not a recommended act (mustahabb); rather, it contradicts the guidance of the Messenger of Allah, peace and blessings be upon him, regarding the optional two rak'ahs before Fajr (Salat Raghibah al-Fajr). He would hasten to pray the Sunnah of Fajr immediately after the adhan concluded. When the muezzin fell silent, he would stand and pray two light rak'ahs before the Fajr prayer, after the dawn had become clear.