All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
As far as we know, there is no religious text indicating that the Sunnah of the Fajr and the obligatory Fajr prayer should be performed successively. Perhaps you are referring to the forbiddance of performing voluntary prayers after the break of dawn. In any case, if you enter the mosque at any time, even during the times when it is forbidden to pray, it is prescribed to perform the two-Rakʻah prayer of Tahiyyat Al-Masjid, because it is a prayer that is performed for a specific reason. The Prophet, sallallahu ‘alayhi wa sallam, said, "If anyone of you enters the mosque, he should not sit until he has prayed two Rakʻahs." [Al-Bukhaari and Muslim]
The forbiddance of performing voluntary prayers after the break of dawn and until sunrise is confined to the voluntary prayers that do not have a specific reason. As for the voluntary prayers that have a reason, the preponderant opinion adopted at Islamweb is that it is permissible to perform them during the times of forbiddance, given the collective evidence in this regard. This is the view adopted by Ash-Shaafiʻi and the one chosen by Shaykh Al-Islam Ibn Taymiyyah .
As for the green dome over the grave of the Prophet, sallallahu ‘alayhi wa sallam, it did not exist during the time of the Prophet, sallallahu ‘alayhi wa sallam; nor that of his Companions; nor that of the first three Muslim generations. Rather, it was built later on, and it has no special virtue.
Allah knows best.