When the congregation begins, there is no Salah but the Fard
قَالَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ إِذَا أُقِيمَتْ الصَّلَاةُ فَلَا صَلَاةَ إِلَّا الْمَكْتُوبَةُ
Abu Hurayrah reported Allah’s Messenger (SAW) as saying, “When the salah (in congregation) stands up, there is no salah except the fard.” [Jami' Tirmidhi: 421 ,Ahmed 8387, Muslim 710, Abu Dawud 1266, Nisai 861, lbn e Majah 1151]
If one misses the sunnah of fajr, he may offer them after fajr
خَرَجَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ فَأُقِيمَتْ الصَّلَاةُ فَصَلَّيْتُ مَعَهُ الصُّبْحَ ثُمَّ انْصَرَفَ النَّبِيُّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ فَوَجَدَنِي أُصَلِّي فَقَالَ مَهْلًا يَا قَيْسُ أَصَلَاتَانِ مَعًا قُلْتُ يَا رَسُولَ اللَّهِ إِنِّي لَمْ أَكُنْ رَكَعْتُ رَكْعَتَيْ الْفَجْرِ قَالَ فَلَا إِذَنْ
Muhammad ibn Ibrahim reported from his grandfather, Qays that as Allah’s Meessenger (SAW) came out, the iqamah of the (fajr) salah was called. So, he offered the prayer with him. Then, the Prophet (SAW) turned back and saw him offer salah. He said, O Qays! Wait! will you pray two salah together? He said, “O Messenger of Allah!
I had not offered the two rakaat (sunnah) of fajr.’ The Prophet (SAW) said, ‘Then there is no harm.” [Jami' Tirmidhi:422, Ahmed : 23821]
About redeeming them after sunrise
قَالَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ مَنْ لَمْ يُصَلِّ رَكْعَتَيْ الْفَجْرِ فَلْيُصَلِّهِمَا بَعْدَ مَا تَطْلُعُ الشَّمْسُ
Abu Hurayrah (RA) reproted that Allah’s Messenger (SAW) said, “He who has not offered the two raka’at of fajr may offer them after sunrise.” [Jami' Tirmidhi: 423]
Also I was able to Find Hadiz on Prophet Praying the 2 Rakath of Dhuhr Prayer after the Asar Prayer
Narrated Kuraib: I was sent to Aisha by Ibn Abbas, Al-Miswar bin Makhrama and 'Abdur-Rahman bin Azhar . They told me to greet her on their behalf and to ask her about the offering of the two Rakat after the 'Asr prayer and to say to her, "We were informed that you offer those two Rakat and we were told that the Prophet had forbidden offering them."
Ibn Abbas said, "I along with 'Umar bin Al-Khattab used to beat the people whenever they offered them." I went to Aisha and told her that message. 'Aisha said, "Go and ask Um Salama about them." So I returned and informed them about her statement. They then told me to go to Um Salama with the same question with which t sent me to 'Aisha. Um Salama replied, "I heard the Prophet forbidding them. Later I saw him offering them immediately after he prayed the 'Asr prayer. He then entered my house at a time when some of the Ansari women from the tribe of Bani Haram were sitting with me, so I sent my slave girl to him having said to her, 'Stand beside him and tell him that Um Salama says to you, "O Allah's Apostle! I have heard you forbidding the offering of these (two Rakat after the 'Asr prayer) but I have seen you offering them." If he waves his hand then wait for him.' The slave girl did that. The Prophet beckoned her with his hand and she waited for him. When he had finished the prayer he said, "O daughter of Bani Umaiya! You have asked me about the two Rakat after the 'Asr prayer. The people of the tribe of 'Abdul-Qais came to me and made me busy and I could not offer the two Rakat after the Zuhr prayer. These (two Rakat that I have just prayed) are for those (missed) ones. [Bukhari :: Hadith 325, Muslim :: Hadith 1815]
Conclusion:
To make the sunnah after Fajr is a provision while doing it on regular basis is a Bid'ah.
The Sunnah is to always pray it before Fajr and no doubt about it. But with a very valid reason, it is okay to pray after sunrise after Fajr
