Are the two Rak'ahs of Fajr prayer considered among those with a cause, such that it is permissible to make them up before sunrise if they were missed due to an excuse, despite the prohibition of praying at this time?

According to the Shafi'is, an expired prayer that is "of a specific cause" (min dhawat al-asbab) is permissible to pray during times when prayer is generally prohibited. For instance, if someone misses the two rak'ahs of Fajr, they are permitted to pray them after the Fajb prayer. This is the madhhab (definition) of the Shafi'is and the Hanbalis, who cite as evidence a narration by Al-Bayhaqi from Qays ibn Sa'd.

As for the Hanafis, they do not perform this qada' (making up of a missed prayer) before sunrise, because it would remain an absolute supererogatory prayer (nafl mutlaq), and absolute supererogatory prayers are disliked after Fajr. Nor do they perform them after sunrise, according to Abu Hanifa and Abu Yusuf. Muhammad ibn al-Hasan (definition), however, disagreed with them, saying that they should be made up after sunrise, and this is also the madhhab of the Malikis. Their evidence is the hadith (definition): "Whoever has not prayed the two rak'ahs of Fajr, let him pray them after the sun rises."