Is it permissible to repeat the Fajr prayer, which was prayed in congregation in the mosque at the time of the false dawn, based on the fatwa of Sheikh Al-Albani?

The criterion is the signs, not unknown calendars. However, one can rely on calendars prepared by trustworthy individuals. Caution is necessary regarding the timing of Fajr, as the call to Fajr prayer in some Arab countries may occur twenty to thirty minutes before true dawn (Fajr al-Sadiq). Consequently, the Dawn prayer (Salat al-Subh) is not permissible at that time, nor does it forbid food for one who is fasting.

If you are in doubt about the validity of the prayer, due to them praying before the true dawn appears, do not pray with them unless you fear being accused. In that case, pray with the intention of a supererogatory prayer, and then pray the obligatory prayer alone afterwards.