Without even getting into the issue of dating of Bhagwad Gita, the ridiculous assertions of the questioner (who seems is more interested in defaming Hinduism than asking questions) will he refuted.
There are dozens of commentaries on the Bhagwad Gita that are older than the 15th Century. Adi Shankara (8th Century) wrote a full commentary on Gita. Similarly Ramanujacharya (11th Century), Madhavacharya (13th Century) have written full commentaries on Gita. Abhinavagupta (10th Century) writes a full commentary on the Gita. Jnaneshwar (13th Century) authored the Jnaneshwari which is a commentary on the Bhagwad Gita. Gitarth Sangraha of Yamunacharya (10th Century) is another full commentary on the Gita. Nimbarka (12th Century) again wrote commentary on Gita. Kadambari by Banbhatta (7th Century) mentions both Gita and Mahabharata in a single Shloka.
Further, all the commentaries post Shankara (and including Shankara) are absolutely consistent with each other to every single Shloka on the Gita. This implies atleast by Shankara's Era (8th Century), there was a single version of Gita available widely from North to South India that was commented upon by all the philosophers. That is the version of the present Gita available
with us dates back to atleast 8th Century CE.
PS. Manuscript dating is never used to identify the date of any literary work due to the loss of older manuscripts, purely due to the fact that preserving bark/paper is extremely difficult, especially in the hot and humid climatic conditions of India. Linguistic analysis of the Sanskrit used in Gita places it not later than 400 BCE. The mention of Gita and Mahabharata together in Kadambari of Banbhatta, which is a purely romantic novel work is enough to conclude that by the 7th Century, Gita as embedded in Mahabharata was an extremely popular text that even playwrights were well versed in.