As I discuss in this question, every Brahmana is a descendant of sages, which is why every Gotra or clan of Brahmins has an associated list of Pravaras, a list of the sages they're descended from. For instance, I am a Brahmin belonging to Vadhula Gotra, and my Pravaras are "Bhargava, Vaitahavya, Savedasa", which means that I am descended from the sage Savedasa, who is a descendant of the sage Vitahavya (whom I discuss here), who is a descendant of the sage Bhrigu. In any case, Gotras often have different branches, so different members of the same Gotra may have Pravaras. This excerpt from a journal paper gives the different sets of Pravaras for Kashyapa Gotra listed in the Baudhayana Shrauta Sutras, a text associated with the Yajur Veda:
i. Kashyapa - Kashyapa-Avatsara-Nidhruva
ii. Rebhas - Kashyapa-Avatsara-Raibhya
iii. Shandila - Kashyapa-Avatsara-Sandila
iv. Laugakshi - Kashyapa-Avatsara-Vasishtha
The first three Pravaras make sense - Nidhruva, Rebhas, and Shandila are all descendants of Kashyapa's son Avatsara. But my question is about the fourth set of Pravaras, "Kashyapa, Avatsara, Vasishtha". This set of Pravaras is also mentioned in this excerpt from the Manava Shrauta Sutras, another text associated with the Yajur Veda.
My question is, why is the sage Vasishtha listed in this set of Pravaras for Kashyapa Gotra? As I discuss in this question, Vasishtha was originally the son of Brahma, but then he was reborn as the son of the gods Mitra and Varuna. Now Mitra and Varuna are Adityas, sons of Kashyapa and Aditi, so Vasishtha would be a descendant of Kashyapa. But Avatsara is a completely different son of Kashyapa than Mitra and Varuna. So what's going on?
I've found one clue that may indicate what's going on. Here is how this excerpt from the Matsya Purana says this set of Pravaras:
I shall now tell you the Rishis of Dvyamushyayana clan, viz. Anasuya, Nakuraya, Snatapa, Rajavartapa, Shaishira, Udavahi, Sairandhri, Raupasevaki, Yamuni, Kadrupingakshi, Jatambi, Divavashta, who have Vatsara, Kashyapa, and Vashishtha for their Pravara.
The word "Dvyamushyayana" means "having two fathers", and it is the standard term used in the context of Niyoga. As I discuss in this answer, Niyoga is the practice where a woman has a man other than her husband father a child with her, either because her husband is unable to father children or because he's dead. The offspring of Niyoga is referred to as Dvyamushyayana, because he has both a biological father and an adopted father. So it looks like Vasishtha and Avatsara are the two fathers in this case.
So are there any scriptures that describe Vasishtha fathering a child with the wife of Kashyapa's son Avatsara? Are there any other scriptures that mention all the descendants of this set of Pravaras given in the Mastya Puranas quote?