According to Bhagavata and Vishnu Purana: Bharata requested Bharadvaja for a son as his he deemed his own progeny unfit to rule. Bharadvaja created "Bhumanyu" out of the sacrificial fire.
An alternate explanation is that "Bhumanyu" was child the Bharata and princess of Kasi.
Either which way, it appears that Bharadvaja presided the ritual of Bhumanyu being adopted by Bharata.
Bhumanyu had five sons: Bfhatkfatra, Jaya, Mahavirya, Nara, Garga. Nara's projeny continued to rule Bharata.
Now it appears that some of these five sons could have joined the Bharadvaja's ashrams to become rishis. These sons continued the lineage of Bharadvaja gotra.
One of the sons Garaga had a son called "Sini" who then had a son called "Gargya". "Gargya" was responsible for establishing the "Gargya" gotra.
In my opinion "Gargya" (son of Sini) could be the famous "Gargya Rishi" who named Krishna. I am basing this on the timelines.
I think there was adoption from Bharadvaja clan to the Bharata clan and peopel from Bharata clan becoming rishis under the Bharadvaja clan.
This lead to Bharadvaja and Gargeya gotra.
In Vedic times there was a Sanskrit convention for forming patronymics: if Garga was the father then Gargi was the son, Gargya the grandson and Gargy-yana the great-grandson.
I think in terms of timeline it makes sense that Gargya rishi was the one who named Krishna and is the progenitor of Gargya gotta.
According to Pravara tradition:
One belonging to Gaargya gotra would say:
Abhivaadaye Aangirasa,(Baarhaspatya) (Bhaaradvaaja), Shainya, Gaargya thrayarsheya
pravaraanvita Gaargya gothrah aapasthamba suthrah
yajuhshaakhaadhyaayii shriiman {your name} sharmaanaamaaham asmi bhoh
||
So the lineage seems to be:
Aangirasa,(Baarhaspatya) (Bhaaradvaaja), Shainya, Gaargya according Pravara.
Sources:
Bhagvata Purana - English Translation: Motilal
Bharata lineage - BORI