According to the Mahabharata, it seems that all the eighteen Puranas were compiled by Rishi Vyasa before the Mahabharata itself was written:
aṣṭādaśapurāṇānāṃ śravaṇād yat phalaṃ bhavet | tat phalaṃ samavāpnoti viṣṇoḥ nātra saṃśayaḥ ||” (MBH 18:06:97)
One devoted to Vishnu acquires (through reading this Mahabharata) that merit which is acquired by listening to the eighteen Puranas. There is no doubt in this.
However, in the Srimad Bhagavatam (which is one of the eighteen Puranas), it is said that Vyasa even after compiling the Vedas and writing the Mahabharata was still sad and Narada visits him to write the Srimad Bhagavatam:
jijñāsitaṁ susampannam api te mahad-adbhutam | kṛtavān bhārataṁ yas tvaṁ sarvārtha-paribṛṁhitam || (SB 1:5:3)
[Narada said]: Your inquiries were full and your studies were also well fulfilled, and there is no doubt that you have prepared a great and wonderful work, the Mahābhārata, which is full of all kinds of Vedic sequences elaborately explained.
śrī-nārada uvāca | bhavatānudita-prāyaṁ yaśo bhagavato ’malam | yenaivāsau na tuṣyeta manye tad darśanaṁ khilam || (SB 1:5:8)
Śrī Nārada said: You have not actually broadcasted the sublime and spotless glories of the Personality of Godhead. That philosophy which does not satisfy the transcendental senses of the Lord is considered worthless.
yathā dharmādayaś cārthā muni-varyānukīrtitāḥ | na tathā vāsudevasya mahimā hy anuvarṇitaḥ || (SB 1:5:9)
Although, great sage, you have very broadly described the four principles beginning with dharma, you have not described the glories of, Vāsudeva.
tvam apy adabhra-śruta viśrutaṁ vibhoḥ | samāpyate yena vidāṁ bubhutsitam | prākhyāhi duḥkhair muhur arditātmanāṁ | saṅkleśa-nirvāṇam uśanti nānyathā || (SB 1:5:40)
Please, therefore, describe the almighty Lord’s activities which you have learned by your vast knowledge of the Vedas, for that will satisfy the hankerings of great learned men and at the same time mitigate the miseries of the masses of common people who are always suffering from material pangs. Indeed, there is no other way to get out of such miseries.
This discrepancy in the timeline of compilation of the Mahabharata and the Srimad Bhagavatam has been often used by Shaktas to claim that Devi Bhagavatam is the true Bhagavata Purana and not the Bhagavatam dedicated to Krishna.
So does it prove the Shakta claim of Srimad Bhagavatam being inauthentic? If not, what is the possible reason for this discrepancy?