All commentators on the Brahmasūtra starts by dealing with the wording of the first sūtra, namely

athāto brahmajijñāsā
Now, after that, there is the desire to know the brahman.

Several topics are discussed in this connection, namely:

• What does “Now” (atha) exactly mean?
• What does “after that” (ataḥ) mean? What does it refer to?
• How should one interpret the compound brahmajijñāsā ‘desire to know the brahman’?
• Why the desiderative in jijñāsā `desire to know’?

As you might imagine, Śaṅkara reads ataḥ as referring only to the study of the Upaniṣads (whereas Rāmānuja reads it as referring to the whole Veda) and denies the role of desire (jijñāsā can in fact also be read as just meaning ‘investigation’).
Veṅkaṭanātha (also known as Vedānta Deśika) decides to start his commentary on the Mīmāṃsāsūtra along the same lines. In fact, UMS 1.1.1 reproduces the style of PMS 1.1.1:

athāto dharmajijñāsā
Now, after that, there is the desire to know dharma

Therefore, he needs to discuss the issue listed above and explains how the study of dharma presupposes the study of the Veda and how desire has a role to play. His discussion of the compound dharmajijñāsā is longer than than the corresponding ones (about brahmajijñāsā) in Śaṅkara and Rāmānuja and displays more familiarity with Pāṇini that I might have hoped for. He explains how the compound must be read as entailing a genitivus objectivus (genitive in the sense of a syntactical object, like when a father speaks of “love of my children” meaning the fact that he loves his children) for several reasons:

dharmasya jijñāsā dharmajijñāsā | karmaṇyevātra ṣaṣṭhī; sambandhasāmānya-
ṣaṣṭhīmaṃgīkṛtya tena tatra karmasāpekṣajijñāsāsāmarthyataḥ karmārthatvaklṛpter viḷambitatvāt | “kartṛkarmaṇoḥ kṛti” iti viśeṣavidhānāt | kartrarthatāyā ihānanvayāt | karmaṇaś cepsitatayā ‘bhyarhitatvāt | anyapare’pi sūtre viṣayaprayojanasūcanopayogāt | ānuṣaṃgikavivakṣāntarasyātrādoṣatvāt | pratipadavidhānaṣaṣṭhīsamāsaniṣedhasya kṛdyogaṣaṣṭhyāṃ pratiprasavena “jño ‘vidarthasya karaṇe” ityādisūtraviṣayatvasthāpanāc ca

The inquiry into dharma is the desire to know dharma. Here the sixth termination denotes the object (genitivus obiectivus), for the following reasons:
1. because, once one has accepted the sixth termination [as meaning] a general connection [“an inquiry related to dharma”], because of that, given that in the [sūtra] an enquiry which [still] requires an object would be impossible, one would be delayed by the fact of [having to] imagine which [referent is implicitly] meant as the object,
2. because of the specific [grammatical] rule “[The sixth case ending is employed after a stem] meaning agent or syntactical object, when [used] along with a kṛt affix” (A 2.3.65), 2.a [and] here (within the sūtra) [the word `dharma’] would not get connected [suitably] as something meaning the agent, 2.b. and because it is proper (abhyarhita“) [that the word `dharma’ denotes the syntactical object], since the object is the most desired [element, as in A 1.4.49],
3. because also in a sūtra dedicated to something else (i.e., to promoting the initiation of the study) it is useful to have an indication (sūcana) of the purpose (prayojana) and of the content (viṣaya) (namely dharma),
4. because there is no fault (doṣa) here of a different intention (vivakṣā) being necessarily implied (ānuṣaṅgika) (i.e., there is no reason to think that the author certainly meant something else),
5. and because it has been established that [the compound dharmajijñāsā interpreted as an objective genitive] is within the range of application of the [group of] sūtras starting with “Of the verb jñā-, when not used in the sense of `to know’, the instrument takes the sixth-case affix” (A 2.3.51) insofar as, in the case of the sixth ending when connected with (yoga) deverbal nouns (kṛt) (such as jijñāsā), the prohibition (niṣedha) of compounds with the sixth ending prescribed for every word* is suspended.

*A 2.2.15 prohibits compounding with nouns ending with the sixth ending if they denote the object (karman). 2.2.16 prohibits compounding with nouns ending with the kṛt suffixes aka and tṛC if they express an agent. 2.2.17 excludes from the prohibition the cases of agents of habitual or professional actions. 2.2.18–22 add further exceptions.

Thus, Veṅkaṭanātha explains that there are good reasons to think that dharmajijñāsa is properly formed as a compound entailing an objective genitive. The reason 5, however, is still puzzling to me. A 2.3.51 regards the usage of jñā– when it does not mean ‘to know’ (but ‘to sacrifice’). Thus, how can it regard compounds such as dharmajijñāsā? However, the sūtras following it all discuss cases of the genitive used in the sense of instrument or object (e.g., of the verbs meaning ‘to remember’). Could Veṅkaṭanātha mean that the group of sūtras starting with A 2.3.51 constitute an exception overruling the prohibition of compounds with the sixth ending, or has 2.3.51 a specific role to play?

On Veṅkaṭanātha’s relation with Śaṅkara’s and Rāmānuja’s models, see my contribution in the book Adaptive Reuse, available here.