Wa Alaykum Assalaam warahmatullaah,
Insha'Allaah you will find the following information useful, though it is important to seek answers from qualified knowledgeable people of Islam, whether that is an Imaam in your area or scholar that you can contact. Unfortunately, we have no such qualified person on this forum to our knowledge, who is available to answer such questions.
With regard to missing prayers, one of the following two scenarios must apply:
1 –
When you miss a prayer unintentionally, for a legitimate shar’i excuse, such as forgetting or sleeping, although you are basically very keen to perform the prayer on time. In this case you are excused but you have to make it up as soon as you remember it. The evidence for that is the hadeeth narrated in Saheeh Muslim (681) which tells how the Prophet (peace and blessings of Allaah be upon him) and his companions slept and missed Fajr prayer, and the Sahaabah began to ask one another, “What is the expiation (kafaarah) for what we have done by neglecting the prayer?” The Messenger of Allaah (peace and blessings of Allaah be upon him) said. “Sleeping does not mean that you are neglecting the prayer. Rather neglecting the prayer means not praying one prayer until the time for the next prayer comes. Whoever does that let him pray when he remembers it.”
This does not mean that a person should sleep deliberately when a prayer is due, until he misses it, then use sleep as an excuse, or neglect a means that would help him to do the prayer, and then take that as an excuse. Rather he must make use of all the means he can, as the Messenger (peace and blessings of Allaah be upon him) did in this case, when he appointed one person to stay awake and wake them up to pray, but that person was overcome by drowsiness, so he did not wake them up. This is the case in which a person may be excused.
From:
http://www.islamqa.com/en/ref/13664
So praying the salaah as soon as you remember would appear to be the best decision, and if you were to pray it after Fajr, that might be delaying it even more.
I also don't know of any reason why a prayer would not be accepted before the start of Fajr. See for example:
Fajr prayer called while praying witr
And Allaah (swt) knows best.