All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, it should be mentioned that the Hadeeth which you mentioned in the question is a fabricated Hadeeth, which means that it is a lie and it is not true that the Prophet said it.
Imaam Ath-Thahabi said about this Hadeeth in his book “Siyar Aa’laam An-Nubalaa’a”: “In my opinion this Hadeeth is fabricated.”
Besides, among the scholars who ruled that it is fabricated is the prominent scholar Al-Albaani in his book “Da’eef At-Tirmithi”.
Anyway, a fabricated Hadeeth is a lie and it is not permissible to act upon it whether in virtuous deeds or otherwise. Also, it is not permissible to attribute it to the Prophet as we have already mentioned.
Moreover, it is an obligation to know that acts of worship are Tawqeefiyyah (i.e. determined by text from the Book of Allaah or the Sunnah of the Prophet and not amenable to personal opinion). So, Allaah is not worshipped except with what He legislated in His Book or on the tongue of His Prophet .
Therefore, it is not permissible for you, dear questioner, to pray that prayer as it is not confirmed in the Sharee’ah as we mentioned.
As regards reciting from the Mus’haf (the Arabic text of the Quran) while performing the prayer, then the Fatwa we adopt here in Islamweb is that this is permissible whether it is done by the Imaam or by a person who is performing the prayer individually. For more benefit, please refer to Fatwa 83531.
Allaah Knows best.