The carbon dating of the Birmingham manuscript, as is evident, is not correct to second/minute/hour/day/year extent, it only provides a range within which it could be possibly located. Rest is to be decided by the inner evidence from the text of the scripture.
The inner evidence rejects that Quran could be before Muhammad's birth for the reasons I have mentioned. Similarly, Muhammad was a real and historic person, the inner evidence of the Birmingham manuscript rejects that Quran descended after Muhammad.
The only possibility is that it was descended on Muhammad within his life span, though codified later in the time of Caliph Uthmān. Birmingham manuscript is similar to Uthmān Codex without change.
One cold observe the correctness of my above expression from the following verse of Quran:
The Holy Quran : Chapter 10: Yunus
[10:16]And when Our clear Signs are recited unto them, those who look not for the meeting with Us say, ‘Bring a Qur’an other than this or change it.’ Say, ‘It is not for me to change it of my own accord. I only follow what is revealed to me. Indeed, I fear, if I disobey my Lord, the punishment of an awful day.’
[10:17]Say, ‘If Allah hadsowilled, I should not have recited it to you nor would He have made it known to you. I have indeed lived among you awholelifetime before this. Will you not then understand?’
[10:18]Who is then more unjust than he who forges a lie against Allah orhewho treats His Signs as lies? Surely, the guilty shall never prosper.
http://www.alislam.org/quran/search2/showChapter.php?ch=10&verse=11
The OP or anybody else to tell us
which other scripture could have come up with the above verse except Muhammad.
Regards