There is no discrepancy at all. The churning of the ocean happened in the current Vaivasvata Manvantara only. It is due to a slight translation error that such a confusion arises.
Let’s look at the verses which begin after Śukadēva Gōswāmī describes demigods of the Cākṣuṣa Manvantara i.e. verse 9 onwards (there are translation errors, which I have marked by Roman numbers):-
तत्रापि देवसम्भूत्यां वैराजस्याभवत् सुतः।
अजितो नाम भगवानंशेन जगतः पतिः॥९॥
tatra api — again in that sixth manvantara; devasambhūtyām — by Devasambhūti(i); vairājasya — by her husband, Vairāja (ii); abhavat — there was(iii); sutaḥ — a son; ajitaḥ nāma — by the name Ajita; bhagavān — the Supreme Personality of Godhead; aṁśena — partially; jagataḥ patiḥ — the master of the universe.
In this sixth manvantara millennium, Lord Viṣṇu, the master of the universe, appeared in His partial expansion. He was begotten by Vairāja in the womb of his wife, Devasambhūti, and His name was Ajita.
Though there is a slight translation error though the gist of this verse remains the same. The correct translation should be-
In that sixth manvantara, Lord Viṣṇu, the master of the universe, in his partial expansion was/became the son of Vairāja in Devasambhūti, named Ajita.
पयोधिं येन निर्मथ्य सुराणां साधिता सुधा।
भ्रममाणोऽम्भसि धृतः कूर्मरूपेण मन्दरः॥१०॥
payodhim — the Ocean of Milk; yena — by whom; nirmathya — by churning(i); surāṇām — of the demigods; sādhitā — produced; sudhā — nectar; bhramamāṇaḥ — moving here and there(ii); ambhasi — within the water; dhṛtaḥ — was staying(iii); kūrma-rūpeṇa — in the form of a tortoise; mandaraḥ — the mountain known as Mandara.
There is a slightly more significant translation error here, and the actual translation should be-
The ocean of milk, having been churned by whom in the form of a tortoise, the nectar of the demigods was received/produced and the moving Mandara mountain was held/ supported.
(Note:- The line was received/produced may not make complete sense in English but translating it into an Indian language like Hindi would make complete sense - क्षीरसगार को जिसके द्वारा मथकर देवों का अमृत प्राप्त हुआ और हिलता हुआ मन्दर कूर्म रूप में धारण किया गया।)
The confusion in interpretation arises because of the words “by whom” used in the second verse. Whether it refers to the Supreme Lord Viṣṇu as a tortoise or Lord Viṣṇu as Ajita in the form of a tortoise, i.e. whether Lord Viṣṇu was born from the womb of Devasambhūti as a tortoise named Ajita.
It clearly refers to the Supreme Lord Viṣṇu (not Ajita) as taking the form of a tortoise (Explained in detail below). The two verses must be combined to read as under:-
Lord Viṣṇu, the master of the universe, by whom the ocean of milk having been churned in the form of a tortoise, the nectar of the gods was produced/received and the moving mount Mandara was held/supported; in the sixth manvantara, in his partial expansion was/became the son of Vairāja in Devasambhūti, named Ajita.
Further, in the next verse king Parīkshita asks as under:-
King Parīkṣit inquired: O great brāhmaṇa, Śukadeva Gosvāmī, why and how did Lord Viṣṇu churn the Ocean of Milk? For what reason did He stay in the water as a tortoise and hold up Mandara Mountain? How did the demigods obtain the nectar, and what other things were produced from the churning of the ocean? Kindly describe all these wonderful activities of the Lord.
In the above verse, the Supreme Lord Viṣṇu is being referred to and not Ajita.
The same becomes clearer when Śukadēva goes onto explain the episode of churning of the ocean-
Verse 17-18 of the very same chapter read as under:
Lord Indra, Varuṇa and the other demigods, seeing their lives in such a state, consulted among themselves, but they could not find any solution. Then all the demigods assembled and went together to the peak of Sumeru Mountain. There, in the assembly of Lord Brahmā, they fell down to offer Lord Brahmā their obeisances, and then they informed him of all the incidents that had taken place.
We can see that Indra and Varuna have been referred to as the demigods. This is a clear reference to the Adityas, as though Indra is a title, Varuna is clearly the name of one of the Adityas.
Thereafter, in the the next chapter, on the command of Lord Viṣṇu, the demigods approach the Asura King Bali-
Then Lord Brahmā and Lord Śiva, after offering their respectful obeisances to the Lord, returned to their abodes. All the demigods then approached Mahārāja Bali. (ŚB 8.6.27)
This King Bali is the very same one, who had been the main reason for whom Lord Vishnu had taken the Vamana Avatara. Meaning he is the son of Virōcana and grandson of Prahlāda. This can be evidenced from verse 29:
The demigods approached Bali Mahārāja, the son of Virocana, and sat down near him. Bali Mahārāja was protected by the commanders of the demons and was most opulent, having conquered the entire universe.
The mention of Bali (grandson of Prahlada and great grandson of Hiranyakashipu) being the leader of the Asuras is indicative of the fact that the churning of the ocean takes place much after Narasimha avatar. As per this answer, the Narasimha Avatar took place in the Vaivasvata Manvanantara only.
This shows us undisputedly that the Indra of the current Manvanatra along with Varuna and other demigods approached the King of the Asuras named Bali in the current Vaivasvata Manvantara only, and therefore clearly indicates the churning of the ocean happened during the Vaivasvata Manvantara.
The final nail in the coffin is hit by the description of Lord Vishnu taking the form of the tortoise in the seventh chapter of the 8th canto itself:
O son of the Pāṇḍu dynasty, when Mandara Mountain was thus being used as a churning rod in the Ocean of Milk, it had no support, and therefore although held by the strong hands of the demigods and demons, it sank into the water. (ŚB 8.7.6)
Because the mountain had been sunk by the strength of providence, the demigods and demons were disappointed, and their faces seemed to shrivel. (ŚB 8.7.7)
Seeing the situation that had been created by the will of the Supreme, the unlimitedly powerful Lord, whose determination is infallible, took the wonderful shape of a tortoise, entered the water, and lifted the great Mandara Mountain. (ŚB 8.7.8)
Thus from the above verses we can see that the Supreme Lord Vishnu took the form of a tortoise only when the demigods and demons were disappointed and the mountain began to sink.
There is no linkage whatsoever with Ajita and the tortoise incarnation. Ajita is one of the forms that the supreme Lord Vishnu took in the previous manvantara. He is the very same Supreme lord Vishnu who in this manvantara took the form of a tortoise.
I will give another example of a translation error in this Srimad Bhagavatam (for finally convincing about the error in translation due to “by whom”)
In ŚB 6.6.40, the translation says that the Asvini Kumaras are the sons of Yamuna (Yami). This is however not true. All sources (even the Amar Chitra Katha) say they were sons of Surya and Samjna, who had taken the form of a horse and mare, in Uttarakuru, respectively. The interpretation error again in this again arises due to the words “she also becoming a mare”. Though the entire verse describes the progeny of Surya and Samjna, the translation has said that Yamuna becomes a mare and begets Ashvini Kumaras. We know that Yamuna as Kalindi weds Krishna and it can’t be true that she’s the mother of the Ashvini Kumaras.
Similarly above the words “by whom” are the confusion creating words.
Thus there is no discrepancy and the churning of the ocean took place in the Vaivasvata Manvanatara, in consonance with the facts stated in Valmiki Ramayana.