As I discuss in this question, by far the most popular school of Hindu philosophy is the Vedanta school, which bases its tenets on the doctrines laid out in the Brahma Sutras, a work by the sage Vyasa that summarizes and systematizes the philosophical teachings of the Upanishads. (You can read the Brahma Sutras here.) But the Vedanta school didn't always have the dominant position in Hindu philosophy; before the time of Adi Shankaracharya the dominant school of Hindu philosophy was the Purva Mimamsa school, which I discuss here. In contrast to the Vedanta school which is devoted to analyzing the Jnana Kanda of the Vedas, i.e. the Upanishads, Purva Mimamsa focuses on analyzing the Karma Kanda of the Vedas, i.e. the Samhitas and Brahmanas.
Now as I discuss in this answer, the Purva Mimamsa school was often borderline agnostic/atheist. But some Purva Mimamsakas, particularly the Purva Mimamsa Acharya Kumarila Bhatta and his followers, believed in the existence of Ishwara or a supreme being, although they did not believe that the Ishwara was responsible for creating the Universe. (They thought the Universe was uncreated and eternal.) But it seems that at least some followers of Kumarila Bhatta believed that Ishwara had a role to play in the attainment of Moksha. Let me explain.
There are two elementary textbooks on Purva Mimamsa which are very similar to each other, Laugakshi Bhaskara's Arthasangraha and Apadeva's Mimamsa Nyaya Prakasha. People have different opinions about which work was written first, but everyone agrees that one of the works was based on the other one. In any case, here is how Laugakshi Bhaskara ends his Artha Sangraha:
Thus we have established that the entire veda which consists of sentences like "he whe is desirous of paradise is to sacrifice etc." immediately or mediately effects duty (dharma) consisting in sacrifices etc. Such (acts of) duty if done with a view to that (result) with a view to which they are enjoined are the cause of that particular result; if done with the intention of making of them an offer ing to God (ishvara) they aro the cause of the highest beatitude. Nor can it be said that there is no authority for performing such acts with the intention of making of them an offering to God; for the following passage of the Bhagavadgita furnishes the required authority "whatever you do, whatever you eat, whatever you sacrifice, whatever you give, whatever you do in way of penance, Kaunntaya, do it all as an offering to me." And the authoritativeness of Smriti is established in the pada treating of Smriti (I. 3.) by the argument of its boing based on Shruti.
And here is how Apadeva ends his Mimamsa Nyaya Prakasha:
And this duty, when it is performed with a view to that with a view to which it is enjoined, produces that (promist fruit). But performed with the intention to offer it to the Exalted Govinda, it produces supreme beatitude. And there is no lack of authority for performing it with the intention of offering it to Him. Because there is the traditional statement: "Whatever tho doest, whatever thou eatest, whatever thou offerest as oblation or givest in gifts, whatever penance thou doest, Son of Kuntï, that do as an offering to Me." And because this (statement) is valid authority, like the smriti-prescription of the Eighth-lunar-day-rite etc. This is set forth in full elsewhere.
Now both Laugakshi Bhaskara and Apadeva were both followers of Kumarila Bhatta. So it seems that at least some followers of Kumarila Bhatta believed that Moksha can be obtained through Nishkama Karma, aka Karma Yoga, where one performs Dharmic actions like Yagnas not for the sake of a material fruit but for the sake of Ishwara. But my question is, did Kumarila Bhatta himself believe that Nishkama Karma is a valid path to Moksha?
Now Kumarila Bhatta wrote three commentaries on the Purva Mimamsa Sutras: the Shloka Vartika, the Tantra Vartika, and the Tuptika. So far I've only found one statement by Kumarila Bhatta concerning methods of getting Moksha, in this excerpt from his Tantra Vartika:
As for the knowledge of Self; both by Conjunction and Disjunction it is found to help the sacrifice ns wel as the Person; because utiliess are knows his self (to be sonnething other than the body that perishes) he would never underfake the sacrifices whose results are said to accrue to the man in another birth. And then again, such passages as-' the Self free from all evil...is to be song.ht after, &c.,” “one should worship the Self" -lay down the knowledge of Self as accomplished by a process accornpanied by due reflection, &c.; and then from such knowledge, we find that there accrues to the agents both kinds of result-Happiness as well as final Deliverance, as is shown by the following passages:-"He obtains all worlds and all desires, passes beyond all sorrow, &c, &c.,"-which speaks of all the eight perfections of Yoga accruing to the person knowing the self; and the passage-“passing his life thus he, after death, reaches the regions of Braima, and from there he never returns'--which points to the attainment of the Supreme Self (Final Deliverance) also as following from a due knowledge of the Seif.
So apparently Kumarila Bhatta believed that Jnana Yoga, i.e. pursuing knowledge of the Atma, is a valid means to Moksha. But does anyone know how he felt about Karma Yoga as a means to Moksha?