What is the ruling on a muezzin dedicating the time between the Adhan of Fajr and the Iqamah to wake people up for prayer, and is that sufficient for him instead of engaging in remembrance (dhikr) during this time?

What this man is doing is lawful and falls under cooperation in righteousness and piety, and the recommendation to awaken a sleeper for prayer. The Prophet, peace and blessings be upon him, awakened Aisha for Witr, and he would not pass by anyone for Fajr prayer except that he would call him or nudge him.

He should also make sure not to miss the pre-Fajr sunnah prayer, and he should avoid raising his voice while talking on the phone so as not to disturb the worshipers.