Very first time our country was named after King Bharata who was born in Priyvrata's lineage in 1st Manvantara (as described in your question). This is supported by Vishnu Purana and Śrīmad-Bhāgavatam (Bhāgavata Purāṇa). Also if we see the below shloka:
उत्तरं यत्समुद्रस्य हिमाद्रेश्चैव दक्षिणम् ।
वर्षं तद् भारतं नाम भारती यत्र संततिः ।।
The country/khand/region (varṣam) that lies north of the ocean and south of the snowy mountains (Himalaya) is called Bhāratam; there dwell the descendants of Bharata. - Vishnu Purana
This is for the king Bharata from 1st Manvantara and not for Shakuntala's son Bharata.
Then why it was renamed?
Point of view 1: It was not renamed actually just those verses are said in praise of the Chakravarati king (Shakuntala's son Bharata) who ruled the entire country. It is also believed that King Bharata (younger brother of Rama) also ruled the entire country (for 14 years). Thats what Lord Shri Rama himself says in Kishkindha Kanda, Valmiki Ramayana.
Point of view 2: There is a Deluge (Jal pralaya) believed in the beginning of current (7th) Manvantra, as described here. And in 7th Manvantra, we don't find any other Chakravarti Samrat named Bharat other that two: Lord Shri Rama's brother and Shakuntala's son. So few scriptures say that our country named as Bharata after them.
But if we consider the Kalpa theory which says that complete annihilation happens after each Kalpa and not after Manvantra. In the beginning of 7th Manvantra was partial pralaya, I mean no reference found for the fresh creation as in the beginning of Kalpa. So Earth was there, the region on which Bharata ruled (Bharata Varsha) was also there.
So point of view 1 seems more appropriate.