Kumārila is an extremely systematic thinker. Thus, if there is a seeming contradiction in Kumārila’s thought, it is likely the case that the contradiction is only a seeming one and that it can be solved.
In the case at stake, we have:

1. Kumārila stating in the ātmavāda chapter within his ŚV that we can directly grasp the self through our awareness of ourselves as an I (via ahampratyaya)
2. Kumārila stating, against Dignāga, that cognitions are not transparent to themselves (see ŚV pratyakṣa 79, 134 and then ŚV śūnyavāda) and that they are rather known a posteriori through arthāpatti based on the fact that something, e.g., a pot, has now the characteristic of being manifest (jñātatā or prākaṭya, see ŚV pratyakṣa 56) and that this characteristic wold be unexplainable otherwise
3. Kumārila stating in the pratyakṣa chapter within the ŚV that perception is sense-perception

Which sort of cognition is ahampratyaya? The “I” cannot be known through cognitions’ awareness of themselves (as in Prabhākara), because of No. 2. Nor can the “I” be known through perception, because of No. 3. Again, given that there must be a solution (the point is too striking to have been missed by Kumārila himself) and that No. 2 looks quite sure, let me try to explore No. 3. Could it be that Kumārila accepts perception as sense-perception AND ahampratyaya? (In this connection, please note that also Jhā writes that “the notion of “I” […] is directly perceptible by the Senses”, fn on ŚV ātmavāda v. 107).

The relevant verses in the chapter on perception are mainly addressed against upholders of yogic perception. Kumārila insists there that perception is about present contents, and the “I” would qualify. He also says that perception arises from a contact between the object and the sense-faculty (sati indriyasaṅgame […] tasya (=pratyakṣasya) evaṃdharmakatvataḥ, vv 17–18), thus excluding svasaṃvedana or other forms of non-sensory direct perception from pratyakṣa. Still, ahampratyaya could qualify as belonging to perception. The sense-faculty at stake in the case of the “I” could be manas, the inner sense faculty. It is in this sense somehow disappointing that Kumārila does not mention the case of ahampratyaya while discussing ātman-manas contact in ŚV pratyakṣa, v. 66. Moreover, ŚV pratyakṣa v. 83 discusses the apprehension of pleasure and pain via manas, but does not mention the case of the manas also cognising the “I”. Even more relevant, v.83 follows a discussion of how a cognition cannot grasp itself and how pleasure etc. can be an exception because there is no svasaṃvedana there, but rather the apprehension of something inhering in the self, via the manas (see Pārthasārathi thereon: manaḥsañjñakenendriyeṇa saṃyuktaḥ ātmā svasamavetān sukhādīn […] pratipadyate). If we were to apply the same scheme to ahampratyaya, we would need to say that the self, connected to the manas, grasps the self itself, which seems to contradict the point made in v. 82, against reflectivity. Also disappointing in this regard are ŚV pratyakṣa, vv. 134–139, that discuss the role of manas in svasaṃvitti (refuted by Kumārila) and pleasure etc. (accepted) and never mention the case of “I-cognitions”. Then, again, ŚV pratyakṣa v. 160 states that manas can work on its own, without an external object, but only cites pleasure and pain as an example (yathā hi manasaḥ sārdhaṃ rūpādau cakṣurādinā | pravṛttiḥ sukhaduḥkhādau kevalasyaiva dṛśyate ||). Same with ŚV śabdanityatādhikaraṇa 337 on manas grasping pleasure etc. when not connected with the external senses.
This is all not conclusive, but it is disheartening that Kumārila never mentions the case of ahampratyaya in the chapter on perception, if ahampratyaya is in fact a case of perception.

At this point I might have been too negative about the possibility of ahampratyaya being a case of perception. Let me therefore look at the evidence in favour of it. The main one is that ahampratyaya is mentioned as an evidence for the existence of the self and as being different than inference in ŚV ātmavāda v. 107. Next, ahampratyaya is discussed in conjunction with recognition (pratyabhijñā) in the same chapter (v. 109, v. 137), and we know that recognition is a combination of memory and perception.

Could it be that ahampratyaya is just a case of recognition? In favour of this view run a few passages in Pārthasārathi’s commentary, where he says pratyabhijñārūpeṇa ahampratyayena (ad ŚV ātmavāda v. 109) and ekasantānasambandhino ‘hampratyayā ekajñātṛviṣayā iti pratyabhijñeti (ad ŚV ātmavāda v. 139), as well as the fact that the self is proven to exist because of phenomena like desire, which need some extension through time (perception of X, memory that X in the past produced pleasure, desire of X, see vv. 104–105). However, recognition requires a preceding (perceptual) awareness (and Kumārila has already established that inference does not establish the ātman). How could one re-cognise something if one had not cognised it in the first place? Recognition is essential in the argumentative structure of ŚV ātmavāda, because this has as its main opponent the Buddhist epistemologists’ point of view on the self being momentary. Yet, it would be strange to claim that ahampratyaya is only a recognition, since this would deny the phenomenological appeal that we recognise ourselves qua-I immediately (not inferentially, as Nyāya-Vaiśeṣikas would like it to be, see vv. 92–101).

At this point I wonder whether for Kumārila ahampratyaya is an ad hoc pramāṇa, a cognition that never ceases every time one cognises something (see ŚV ātmavāda 133), but different than the arthāpatti through which one knows a posteriori that a cognition has taken place. Now, against that runs Kumārila’s mention of six pramāṇas (e.g., ŚV codanā 111cd), which seem to either point to ahampratyaya as a subspecies within one of the main sources of knowledge or to ahampratyaya as not necessarily knowledge-conducive. Could be just a cognition (jñāna), one that is not necessarily true, but that remains valid as long as it is not falsified? However, a pramāṇa is in Kumārila exactly defined as something that remains valid as long as it is not falsified. That is, in Nyāya it is easy to distinguish pramāṇas from thinking processes that are not *necessarily* knowledge-conducive although they might lead to true cognitions, such as tarka. But this distinction does not seem to be available to Kumārila’s falsificationism. He does use the term pramāṇa also for sources other than the six (pratyakṣa, anumāna, śabda, upamāna, arthāpatti, abhāva), for instance in the case of pratyabhijñā and of the hermeneutic pramāṇas, but both can be reduced to a subspecies of one of the six pramāṇas. This all would lead one back to ahampratyaya being a form of manas-pratyakṣa, with the ātman featuring both as the subject and object thereof.

ŚV ātmavāda 142 seems to offer a slightly different point of view, with the ātman being described as self-luminous (through the simile of a light, jyotiḥ). Thus, even though Kumārila refutes (see point No. 2 above) self-luminosity in the case of cognitions, he seems to accept it for the self. The next verses explains that this does not apply to other people’s selves, that need to be known through inference (v. 145) and perhaps also through analogy and linguistic communication combined (v. 144). Unfortunately, Uṃveka’s commentary is not available for this chapter and Pārthsārathi does not add much to this short section.

By the way, one might wonder whether ahampratyaya could ever count as knowledge, given that it is unfalsifiable. However,
a. Kumārila has defeated the alternatives already (mainly bhūtacaitanyavāda, or physicalism and the non-physicalist illusionism of Buddhist epistemologists)
b. Kumārila showed at least in one case that he is comfortable with an unfalsifiable knowledge, namely the Veda.