Wa`alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother, many thanks for your trust in us and your keenness to make up for your missed Prayers that we ask Almighty Allah to accept from you and forgive your negligence in performing missed Prayers.
It is disliked to pray optional prayers after `Asr or Fajr Prayers. However, making up for missed Prayers, praying the Funeral Prayer and some other Prayers that have a specific reason is permissible during these times.
In his response to your question, Dr. Main K. Al-Qudah, a member of AMJA Permanent FATWA Committee, stated:
The prohibition of praying after Fajr and `Asr is limited to praying only optional prayers which have no specific reason, but if there is a reason-like in your case- then praying after Fajr and `Asr is allowed.
It was authentically narrated that the Prophet (peace and blessings be upon him) himself prayed two rak`ahs after `Asr. And, when he was asked why, he replied: "I am making up for the ZhuhrPrayer Sunnah that I missed due to another duty I was involved in." (Al-Bukhari and Muslim)
Based on the above, continue making up for all that you missed in the past until you are sure you have made up for all the missing Prayers.
May Allah help you in doing so, increase your knowledge, and assist you in applying what you learn.