Since no one has yet replied to this, perhaps there is not much new on this question. As we all know, the continuing upabṛṃhaṇa, expansion or supplementation, of the purāṇas makes them almost impossible to date accurately. Since their older parts may be separated from their later parts by centuries, are we seeking the date of their latest redaction?
In the critical edition, The Bhāgavata (ed. H. G. Shastri, Bharati K. Shelat, K. K. Shastree, B.J. Institute of Learning and Research, Ahmedabad, 1996-1999), an "Epilogue" volume was published in 2002 (vol. 4, part 3), which has just over two pages on "Time and Composition of the BGP" (pp. 15-17). K. K. Shastree there concludes (p. 16): "It means that the time of composing the BGP is not earlier than the latter half of the 7th cent., A.D. and not later than the first half of the 8th cent., A.D."
However, Ganesh Vasudeo Tagare in the Introduction to vol. 1 of his English translation of The Bhāgavata-purāṇa (1976, p. xxxvi) points out that a Purāṇa Bhāgavata is named in the Jaina text, Nandī-sūtra. Since this Jaina text can be dated around 500 C.E., some version of the Bhāgavata-purāṇa existed then. This is in his section, "The Date and Authorship of the Bh.P.," pp. xxxiv-xlii. He begins this section by listing thirteen proposed dates for the Bhāgavata-purāṇa, ranging from 1200 B.C. to 1300 A.D., and their sources. Included here is the date proposed by purāṇa specialist R. C. Hazra, author of the 1940 classic work, Studies in the Purānic Records on Hindu Rites and Customs, as 600 A.D.