How do Buddhists reconcile "Anatta" with Buddha supporting the existence of the Self in the Mahayana Mahaparininirvana Sutra?

I am not an expert on Mahayana Buddhism, but I will try to answer this question.

I found another translation here.

For the same section quoted by OP, this other translation states:

Bhikṣus, thus you should not cultivate your ideas of impermanence, suffering, nonself, impurity, and so forth as if they were the true meaning [of the human condition]. Just like those people who took rocks, vegetation, or gravel to be the jeweled necklace they were looking for, you must carefully learn about expedient means so that wherever you are, you can continually cultivate your perception of self as well as your perceptions of permanence, bliss, and purity.

Later on, the text says:

And out of my desire to subdue the non-Buddhist paths I therefore declared: “There is no self,there is no person or [individual] living being, life span, personality, observer, actor, or experiencer.” Monks, the heterodox paths affirm a “self” in the same manner as [some infer literacy in] the shapes of letters incised into wood accidentally by insects. This is why the Tathāgata proclaims “nonself” as part of his buddha-dharma. It is because I need to straighten out [the thinking] of living beings—because I am aware of their situation—that I expound the absence of self. .... But what I am speaking of is not what ordinary people imagine the self to be. Ordinary people or ignorant people suppose the self to be the size of a thumb, or perhaps a mustard seed, or a speck of dust. What the Tathāgata explains the self to be is nothing like that. Therefore when I preach “dharmas are without self,” in truth they are not without self. So what is this self [of which I now speak]? If a dharma is true, real, permanent, autonomous, a basis, and its nature is immutable, then that is what I call self.

The term "dharma" above means "thing". So "self" is a thing which is true, real, permanent, autonomous, a basis and is immutable.

Then it continues:

[Kāśyapa continued:] “World-Honored One, is there a self or not in any of the twenty-five forms of existence?”

The Buddha said: Good man, “self” is precisely what tathāgatagarbha means. All living beings have buddha-nature, and this is what is meant by this notion of self. However, the significance of “self” understood in this way has been continuously covered over by an uncountable number of the defilements since the beginning [of any given individual’s existence], and that is why living beings have been unable to perceive it.

So, the "self" in the Mahayana Mahaparinirvana Sutra refers to the tathagatha-garbha or Buddha-nature.

It must be noted that tathagatha-garbha or Buddha-nature is not the same as the concept of Atman in Hinduism, as can be seen by the following excerpt from the same text Mahayana Mahaparinirvana Sutra:

Good man, if people have never heard of the profound, hidden tathā- gata treasury, how could they be aware of the existence of buddha-nature? What is it that I am calling “a hidden treasury”? I am talking about the well-balanced Mahāyāna scriptures themselves. Good man, there are other paths: some expound a self that is permanent, some expound a self that is cut off permanently [at death]. The Tathāgata is not like them. In affirming both self and nonself, I call it “the middle path.” One way of explaining this is that the Buddha expounds a middle path in which all living beings possess buddha-nature, but because it is obscured by the defilements they do not understand it and do not see it. Therefore you must diligently cultivate whatever expedient means you can in order to destroy those defilements.

According to this article:

Tathagatagarbha, or Tathagata-garbha, means "womb" (garbha) of Buddha (Tathagata). This refers to a Mahayana Buddhist doctrine that Buddha Nature is within all beings. Because this is so, all beings may realize enlightenment. Tathagatagarbha often is described as a seed, embryo or potentiality within each individual to be developed.

And it discusses further:

In the religions of the Buddha's day that were the forerunners of today's Hinduism, one of the central beliefs as (and is) the doctrine of atman. Atman means "breath" or "spirit," and it refers to a soul or individual essence of self. Another is the teaching of Brahman, which is understood as something like the absolute reality or the ground of being. In the several traditions of Hinduism, the precise relationship of atman to Brahman varies, but they could be understood as the small, individual self and the big, universal self.

However, the Buddha specifically rejected this teaching. The doctrine of anatman, which he articulated many times, is a direct refutation of atman.

Through the centuries many have accused the Tathagatagarbha doctrine of being an attempt to sneak an atman back into Buddhism by another name.

In this case, the potentiality or Buddha-seed within each being is compared to atman, and Buddha Nature -- which is sometimes identified with the dharmakaya -- is compared to Brahman.

You can find many Buddhist teachers speaking of small mind and big mind, or small self and big self. What they mean may not be exactly like the atman and Brahman of Vedanta, but it's common for people to understand them that way. Understanding Tathagatagarbha this way, however, would violate basic Buddhist teaching.

Another article here tries to point the origins of tathagatha-garbha or Buddha-nature to the Luminous Mind in the Pali Canon (which according to Theravada Buddhism, is not permanent or eternal, but is dependently arising - see this answer):

The term "'tathagatagarbha'" is generally taken as to mean that the "garbha" (womb or potential) of a 'Tathagata' exists in all sentient beings without exception, and though temporarily contaminated by adventitious defilement ('agantukaklesa'), it is the cause which eventually leads sentient beings to enlightenment. The notion of the 'tathagatagarbha' can be traced to a luminous¡A inherently pure mind (pabhassar citta) found in the 'Anguttara-nikaya' (1:5):

Oh! 'Bhiksus'. The mind is pure! It is defiled by the adventitious defilement. Oh! 'Bhiksus'. The mind is pure! it obtains liberation through the adventitious defilement.

When the original pure mind came to be regarded as something capable of growing into Buddhahood, there was the 'tathagatagarbha' doctrine. Although the concept of an intrinsically pure mind exists in the Nikaya Buddhism, many Buddhologists, such as Wayman (1), Paul (2), Yin-shun (3) think that the 'tathagatagarbha' thought was originated from the 'Mahasamgika', but was rejected by the 'Theravada'. This theory is also held by Mizuno who points out that the pure mind ('pabhassarcitta') articulated in the Nikaya Buddhism is not totally identical with the original pure mind ('prakrtivisuddhi-citta') articulated in the 'Tathagatagarbha' doctrine, for Mizuno asserts that the former is static whereas the latter is dynamic in that it is capable of eradicating defilement.(4) At any rate, the relationship between pure mind and the adventitious defilement appears to have been wholly adopted by the 'Mahasamghika' and later by the 'Mahayana'.

It also comments in its conclusion:

In conclusion, when we try to interpret the thought of the 'tathagatagarbha', we should keep several points in mind:

1) The 'tathagatagarbha' symbolizes the potential for enlightenment (a principle) rather than a material "essence" of ultimate truth,

2) the 'tathagatagarbha' is based on the framework of the 'Mahayana' doctrine of 'sunyata-pratitya-samutpada'.

3) The development of the 'tathagatagarbha' doctrine signifies the ability of a religious tradition to meet the spiritual needs of the masses aiming at a given time.

That is to say the 'tathagatagarbha' thought was formed as an positive soterio-logical approach to counteract the "'sunyam sarvam'" (all is empty) view. The 'tathagatagarbha' which strongly articulates a devotional and experiential approach to salvation provides much to the hope and aspiration of the people at large. It is this positive aspect that was taken up and strongly emphasized in Chinese Buddhism.

4) The 'tathagatagarbha' doctrine is employed as a skill-in-means ('upaya'). This does not necessarily mean that the theory of the 'tathagatagarbha' is neyartha, a teaching requiring further qualifications -- rather, it is a skill-in-means in that it is taught to suit the needs of a certain kind of people and circumstances. This is why it is said in the 'sutra' that in order to teach the emptiness of all dharmas, the Buddhas preach sometimes by the doctrine of the 'tathagatagarbha', and sometimes by that of emptiness. Thus it is better to take the 'tathagatagarbha / Buddha nature' as representing "profound existence" derived from "true emptiness" rather than as a monistic self.