Is there any problem if I delay going to the mosques near me to join them in the second rak'ah after a third of an hour has passed, knowing that they establish the Fajr al-Sadiq prayer early?

The time for Fajr prayer begins with the appearance of the true dawn (al-Fajr al-Sadiq). One who cannot discern the time of Fajr themselves should rely on a trustworthy muezzin or an approved timetable. If the mosque relies on a reliable timetable or a muezzin who is knowledgeable of the prayer times, there is no sin if the worshipper joins the prayer in the second rak'ah. However, if the worshipper doubts whether the time has entered, and there is no approved timetable or knowledgeable muezzin, their prayer is invalid until they are certain or strongly presume that the time has entered, because doubt about the entry of the time invalidates the prayer. If the entry of the time is confirmed or strongly presumed, and the Imam is in the second rak'ah, it is permissible to join him.