As I discuss in my question here, Vishnu is said to have taught the principles of Vaishnavism to four disciples: Lakshmi, Brahma, Shiva and Sanatkumara. And in turn they started the four main Sampradayas or traditions of Vaishnavism: Sri Sampradayam, Brahma Sampradayam, Rudra Sampradayam, and Kumara Sampradayam. My question is about the Rudra Sampradayam, the one founded by Shiva. Currently the only Vaishnava sect which traces its origins to the Rudra Sampradayam is Pushtimarga, the sect of Vallabhacharya which worships the Srinathji form of Vishnu (which I discuss here.)
Now Vallabhacharya's followers believe that Vallabhacharya belonged to the Guru Parampara of Vishnuswami, an earlier Acharya of the Rudra Samparadayam. But the Gaudiya Vaishnava sect, which traces its origins to the Brahma Sampradaya, claims that before Vallabhacharya joined Vishnuswami's sect, he was originally a Gaudiya Vaishnava. Here is what Srila Bhaktivedanta Prabhupada, the founder of the popular Gaudiya Vaishnava organization ISKCON, says in one of his commentaries:
Vallabha Bhaṭṭa was a great learned scholar of Vaiṣṇavism. In the beginning he was very much devoted to Śrī Caitanya Mahāprabhu, but since he thought that he could not receive proper respect from Him, he later joined the Viṣṇu Svāmī sect and became the ācārya of that sect. His sect is celebrated as the Vallabhācārya-sampradāya. This sampradāya has had great influence in Vṛndāvana near Gokula and in Bombay.
And Vallabhacharya is mentioned in two chapters of the Chaitanya Charitamrita, the traditional biography of Chaitanya Mahaparbhu, the founder of Gaudiya Vaishnavism. In this chapter, Vallabhacharya meets Chaitanya Mahaprabhu at the confluence of the Ganga and Yamuna rivers, has religious discussion with Chaitanya Mahaprabhu, ferries him across the Yamuna river to his house, serves him devotedly in his house, and then ferries him across the Ganga river to Prayaga.
In another chapter of the Chaitanya Charitamrita, Vallabhacharya goes to Puri to meet Chaitanya Mahaprabhu and gives food to Chaitanya Mahaprabhu and his associates. Then some time later, Vallabhacharya comes back and tells Chaitanya Mahaprabhu that he has just written a commentary on the Srimad Bhagavatam, and that it's superior to Sridharaswami's commentary. Chaitanya Mashaprabhu takes offense to this, as he has great respect for Sridharaswami's work. This leads Vallabhacharya to lose his false pride, and he ultimately receives initiation from Chaitanya Mahaprabhu's shishya and childhood friend Gadadhara Pandita.
My question is, do Vallabhacharya's followers agree that Vallabhacharya was originally a Gaudiya Vaishnava? How do they characterize the encounters between Vallabhacharya and Chaitanya Mahaprabhu, assuming they even happened? And do they accept that Vallabhacharya received initiation from Chaitanya Mahaprabhu's shishya Gadadhara Pandita?