Bhagavat Purana is a ‘smriti’ literature and that means it would have been (might have been) updated few times.
Thus the timing we are going explore could be the timing of its first composition or that of one of the updates.
At a minimum, it means that we have ‘testable’ evidence of either its composition or one of the updates that takes us back to anywhere from 750 BCE through 1900 BCE.
Let’s look at the evidence:
The period so defined is broad because the composer/updater of this portion of Bhagavat Purana, chose to employ zodiacs instead of nakshatra in stating the positions of equinoxes and thus our ability to pinpoint the timing of this event has error of +/- 15 degrees (Zodiacs) vs. +/- 6 degrees (nakshatra).
The point of spring equinox resided in the region of Mesh rashi (Aries) from 2000 BCE through 50 BCE.
The point of fall equinox resided in the region of Tula rashi (Libra) from 2300 BCE through 700 BCE.
Thus overlapping time interval runs from 2000 BCE through 700 BCE.
Further narrowing of this time frame can be done by aligning points of equinoxes with the corresponding YogaTaras (yogatara(s) of Vishakha and Yogatara of Bharani) within the overlapping time frame and that leads to either a time period of 1200 BCE (based on Yogatara of Bharani) or to a time period of 1300 BCE or 1800 BCE (based on Yogatara(s) of Vishakha).
This would mean, we may estimate that the timing of composition or one of the updates of Bhagavat Purana was about 1200/1300 BCE. Do keep in mind that someone looking at the same data and doing ‘back of the envelope’ calculations may estimate the timing that would vary from 1200/1300 BCE by few centuries.
For example, see here:
We would need additional and independent evidence to refine this timeframe further.