Question
Can we date the Bhagavad-Gita?
Answer

Generally, scholars claim that Buddhism is older than the Gita because the Gita uses the word nirvāṇa five times. Several years ago, I sent the following message to the main academic Indology discussion group, called Indology, with leading scholars from the best universities:
There are three logical possibilities:
1. The Gita borrowed the term nirvāṇa from Buddhism
2. Buddhism borrowed it from the Gita
3. Buddhism and the Gita are mutually independent in their use of the term.
I then asked for solid evidence that #1 is the case, that the Gita borrowed the term from Buddhism.
Amazingly, not one scholar in the world had an argument to support #1, yet practically all of them teach that the Gita borrowed the term from Buddhism, and thus came later. I repeat, not one scholar could furnish an argument to support what they all teach.
There are many such cases in academia.
Before explaining the Gita’s antiquity, I believe we should start with an explanation of a roughly 5000 year old date for Kṛṣṇa’s life on Earth. The best evidence (and perhaps the source of the 5000 number] still seems to be the archeo-astronomical calculations of the great Arya-bhata, a co-founder of trigonometry, and exceptional astronomer [476 AD to 550 AD].
Anecdotally, it would be very interesting to study all that we know of the Earth 5000 years ago, which is the division of yugas. There is some evidence that major changes took place around that time. Here are just a couple that come to mind:
1. Earth weather changed. The El Niño effect began around 5000 years ago.
2. There were major dynastic shifts in Mesopotamia around 5000 years ago. This would be a natural occurence in a post Kurukṣetra world.
One can make separate calculations for the date of the text as we now have it, and the date of the events themselves.
We have hardly begun to address this issue, so let us not give up hope yet of formulating a reasonable justification for our 5000 year claims. One can make separate calculations for the date of the text as we now have it, and the date of the events themselves.
It seems there is a tendency among some scholars to conflate these two dates, or to peg the latter on the former, which of course is dangerous. There will always be a coterie of scholars who resist great antiquity, for their own reasons.
We would have to see if there arguments for later dates, in opposition to Aryabhata.