The possible explanation of your question is provided by author of the book Vedanta-Darshan (Brahma Sutra) Harikrishna Das Goyandaka by Gita Press.
In his introduction of the book , the author is precisely talking about your question and the doubts regarding this matter. According to the author ,there is no problem in believing that the brahma sutras were authored by Ved -Vyasa whoes other name is also Badarayana. Below is the english translation of the explanation from the book.
Herewith i would like to discuss some explanatory things regarding
Brahma-Sutras.This is (Brahma-Sutras) is very ancient text. Some
scholars think that its a fairly new one , by looking at the
criticism of Samkhya , Vaisheshika ,Buddha, Jaina , Pashupata and
Pancharatra. And do think that Badrayana is different from Ved-Vyasa.
But this is their false notion. The philosophies which are criticized
in Brahma-Sutras are the eternal opinions /philosophies/views are
eternal which are in flow i.e. those views are timeless . The great
debate between SadVada and AsatVada(Astika and Nastika) is going on
since from the ancient Vedic time itself. The Sutrakara not mentioned
the Acharyas of these philosophies anywhere in his work. He only
reviewed "PradhanKaranVada(Samkhya) , AnukaranVada( vaisheshika) ,
VigyanVada etc. The word Badrayana is associated with Ved-Vyasa since
long and used to mention one person only. Panini also mentioned the
name of the text Bikshu-Sutras by son of the Parashara called Vyasa
Key Points -
- Samkhya , Vaisheshika ,Boudha, Jaina , Pashupata and Pancharatra were eternal thoughts /views. They later formed their respective schools , but initially they were views only , before they got their seprate exsistance.
- Bikshu-Sutras was the one of the name of Brahma-Sutras written by Ved-Vyasa (son of parashara muni) ,which is mentioned by Panini .
- Ved-Vyasa OR Badarayana only reviewed these thought /view systems and not the religions or philosophical schools.
This proves that Brahma sutras were written By Sage Rishi Ved Vyasa , whoes another name is Badarayana. And they both are same. And Buddhism and Jainism etc. mentioned /criticized were views /thoughts ,which are present from the beginning i.e. Astika VS Nastika.
Here is the screenshot
of the above introduction in Hindi.
In Puranas also Shree Ved-Vyasa is called as a Badrayana. For example Shreemad Bhagvat Purana calling Rishi Ved-Vyasa as Badaryana in the following Shloka.
यानि वेदविदां श्रेष्ठो भगवान् बादरायण: ।
अन्ये च मुनय: सूत
परावरविदो विदु: ॥ SB 1.1.7 ॥
yāni veda-vidāḿ śreṣṭho bhagavān bādarāyaṇaḥ
munayaḥ sūta parāvara-vido viduḥ
O suta ! Whatever the venerable Badrayana greatest among the learned
ones ,knows and whatever other sages who knows saguna and nirguna
Now see the footnote clearing declaring Ved-Vyasa the author of Brahma Sutras is same as Badrayana. see p.19.