Is it permissible for me to pray Fajr in a small mosque whose imam follows the Sunnah and prays with tranquility, or in a larger mosque whose imam does not follow the Sunnah and prays quickly?

There is no harm in praying in either the large or the small mosque, and it is not obligatory to pray in the large mosque. If praying in the small mosque is more conducive to khushu' (devotional humility) and more in line with the Sunnah, then it is better; because the virtue related to the prayer itself takes precedence over that related to its location. Jurists have discussed these matters from the perspective of preference, not obligation.