Almost everything narrative that Hindus know about Krishna's childhood and youth comes from the Bhagavata Purana, and almost nothing about when it was written is agreed.
The general problem is structural. Ludo Rocher put it flatly: it is not possible to set a specific date for any Purana as a whole, and proposed dates continue to vary widely and endlessly. Cornelia Dimmitt and J.A.B. van Buitenen noted that the manuscripts are encyclopedic and that it is difficult to establish when, where, why or by whom they were written. The reason is that a Purana is not a book that was composed once. It has an oral prehistory, it accreted material over centuries, and it was revised by people who did not mark their revisions. Asking for its date is asking a question the object is not shaped to answer.
For the Bhagavata specifically, modern scholarship generally places it between the sixth and the tenth centuries, with many putting the final redaction in the ninth or early tenth. Sheridan argued it was composed by learned brahmin ascetics, probably in South India, and shows the influence of the Tamil Alvar poets. Two things pull in opposite directions. Vedic archaisms and pre-Gupta dynastic lists pull the date early, and Edwin Bryant and D. Dennis Hudson have argued for the Gupta period, between the fourth and seventh centuries, Hudson on iconographic grounds from the Vaikuntha Perumal temple. Against that, the tenth- and eleventh-century South Indian Vaishnava theologians Yamuna and Ramanuja do not cite it, a silence that is hard to explain if the text was already authoritative.
Wendy Doniger's approximate datings for other Puranas give a sense of the spread: Markandeya around 250, Matsya between 250 and 500, Vayu around 350, Vishnu around 450, Linga anywhere from 600 to 1000. None of these is a publication date. They are estimates of when a text reached something like the shape it now has.