Is it obligatory to join the congregation in mosques that pray Fajr early, even if one misses a rak'ah, or is it better to wait for mosques that delay the prayer until just before sunrise, knowing that I am unable to perform the Sunnah prayer before Fajr in early mosques?

It is preferable to pray Fajr at the beginning of its time, i.e., 30 minutes after the Adhan (definition), as long as the time has begun. If there are two mosques in the neighborhood, one that prays early and another that delays, it is better to pray in the mosque that delays in order to catch the opening Takbeer (Takbeerat al-Ihram (definition)). This is permissible if the two mosques are in the same neighborhood. However, if the other mosque is outside the neighborhood, then according to Imam Ahmad, it is better not to move to it, while Hishim permitted it. Generally, there is flexibility in the matter, and it is beneficial to move to another mosque to catch the supererogatory prayer (nafilah). One should also go early to catch the Fajr Sunnah (definition) prayer in its proper time.