The one-mind is not a thing, but exists in the way that patterns and laws exist. Like, Newton's Laws of Thermodynamics are not existent entities, but it would also be incorrect to say that they don't exist, you know? One-mind is likewise not meant to be interpreted as an entity.

Sallie King explains this very well in her monograph on Vasubandhu's Buddha Nature Treatise, also translated by Paramartha, and helps together with the Awakening of the Faith to demonstrate Paramartha's understanding of Yogacara thought.

One-mind in English is a very tough concept to grasp, a large part due to how it's constructed and translated. Even the way that westerners type it out as "One Mind" accidentally reifies it into a substantialist entity, whereas the one-mind really refers to a unified quality that is revealed when the obscurations of mind have fallen away, which all beings possess.

This kind of emptiness is not nothingness, and not an absence of substance, but the field upon which the dialectic of being and non-being plays out. This is what gives it a "generative" power and why it is rendered into a positivist expression in East Asian thought. But this is still very much a theory of emptiness, just expressed through East Asian logic (or Indian logic re-expressed through East Asian logic). Unfortunately, the Awakening of the Faith is very terse and does not completely make the logical argument for its position. That is why Paramartha also translated the Buddha-nature Treatise, in order to showcase the complete view that is being presented.

Unlike emptiness teachings, this One Mind truly exists and is not empty

Are you sure about this?

how is Suchness endowed with Buddha qualities still compatible with emptiness teachings

How is it incompatible?

The One-Mind is the totality of the Dharmadhatu, that is the unity of emptiness and existence. That is what ‘One’ refers to. Not numerical ‘one’, but rather the Union and non-duality of existence and emptiness. It is referred to as “Mind”, as it naturally possesses an innate awareness, distinguishing it from rocks or dirt.

There can be no emptiness apart from the One Mind, how could it not be compatible with emptiness? Remember, emptiness does not mean there is nothing at all. It is the negation of our deluded imputation of svabhava or the self-existence of particular entities.

What the Awakening of Faith and other Tathagatagarbha texts are describing, is the nature of realty, suchness, once it has been freed from our conceptual proliferation. That is reality free from our delusional imputation - reality that is empty.

To use a metaphor, emptiness is telling us the images that appear inside a tv aren’t real. No particular person or thing in the TV has any true existence, even if we think they do. The One Mind is that TV itself. But of course, unlike a TV, the One Mind isn’t any particular thing but the totality of depend origination.

Emptiness inability to locate any particular entities within the net of dependent origination. As Nagarjuna says, “whatever arises, that I say is empty”. Emptiness is to be found in dependent arising.

The Awakening of Faith, identifies dependent origination to be linked to the mind, since it is only possible for mind to construct the appearances of arising, in the absence of it, for individual phenomena to be created in the absence of them. Like when we gaze up in the sky and imagine shapes or pictures. Those do not belong the sky but the mind.

So this One Mind, is the totality of dependent origination, that is empty. Free from any delusional proliferation. Yet it is not non-existent. Although speaking ultimately there isn’t any single phenomena to be identified within, it still conventionally contains all positive qualities of awakening.

After-all, the Buddha didn’t puff out of existence after awakening. He continues to liberate all sentient beings, he possesses the three bodies, four wisdoms, various Purelands, and so on.

The “one mind” being used in reference to dharmakāya has to do with dharmakāya being a generic characteristic (samanyalakṣana).

The dharmakāya is not a thing, but rather a realization. Since all Buddhas realize the same thing about the nature of mind and phenomena, we cannot really say there are different dharmakāyas, hence the idea of it being “one.” However it is not literally “one” like the monistic natures taught in non-Buddhist teachings.

Classically, dharmakāya is described as “neither one nor many.” It isn’t “one” because various Buddhas realize it independently. And also is not “many” because it is uniform in characteristic wherever it is realized. Like the heat of a flame is not “one” because there are distinct instances of fire, yet not “many” because fire is the same in characteristic wherever it is found.

For example Asaṅga says in the Mahāyānasaṃgraha:

[Likewise, the dharmakāya has] the characteristic of the nonduality of oneness and difference because the tathāgatagarbha is not different [in characteristic], while innumerable [distinct] mindstreams [of bodhisattvas] reach fully perfect awakening.

Everything in Samsara is empty (dependently originated). But the One Mind is the luminous, pure mind observing all things, the unconditioned outside Samsara, the true reality beyond the illusion.

In the Vimilakirti Sutra and Lotus Sutra, the Buddha describes how our world is a Pure Land.

Then when their minds are filled with yearning, at last I appear and preach the Law for them. Such are my transcendental powers. For asamkhya kalpas constantly I have dwelled on Holy Eagle Peak and in various other places. When living beings witness the end of a kalpa and all is consumed in a great fire, this, my land, remains safe and tranquil, constantly filled with heavenly and human beings.

• Lifespan of the Thus Come One, Lotus Sutra

I don't think I understand how this is incompatible with the doctrine of emptiness.

As Nagarjuna formulated emptiness, it was that all phenomena is interconnected, and there is no intrinsic property because all properties arise from various causes and conditions. This was later expounded upon by T'ien-t'ai's doctrine of three thousand realms in a single moment of life. The 10 worlds are mutually inclusive, and all contain the 10 factors of life across the three existences. This is to say that in each moment, the potential for manifesting one's Buddha nature exists along side with all the other worlds. Buddhahood, hell, and heaven are not mutually exclusive.

If this is true, how is Suchness endowed with Buddha qualities still compatible with emptiness teachings?

Rangtong is what is found within conditions.

Shentong is what is known free of those conditions.

The realization of buddhahood is the realization of the underlying unconditioned state (see the nibbanadhatu sutta) via the cessation of conditions such as occurred under the bodhi tree.

It is the realization of the emptiness of any independent causation or origination to be found in any condition that can ever be known. 

They all collapse back into this underlying pure awareness. 

There is only one resolution-self-sprung awareness itself, which is spaciousness without beginning or end; everything is complete, all structure dissolved, all experience abiding in the heart of reality.

So experience of inner and outer, mind and its field, nirvana and samsara, free of constructs differentiating the gross and the subtle, is resolved in the sky-like, utterly empty field of reality.

And if pure mind is scrutinized, it is nothing at all it never came into being, has no location, and has no variation in space or time, it is ineffable, even beyond symbolic indication and through resolution in the matrix of the dynamic of rigpa, which supersedes the intellect-no-mind! nothing can be indicated as "this" or "that," and language cannot embrace it.

In the super-matrix-unstructured, nameless all experience of samsara and nirvana is resolved; in the super-matrix of unborn empty rigpa all distinct experiences of rigpa are resolved; in the super-matrix beyond knowledge and ignorance all experience of pure mind is resolved; in the super-matrix where there is no transition or change all experience, utterly empty, completely empty, is resolved.

~Longchenpa

That is the truth body of a buddha, the perfected mode of reality, the heart of the tathagatagarbha as described in the heart sutra.

The sambhogakaya and nirmanakaya are the results of the accumulation of the repository consciousness.

But they are not what a buddha realizes as their truth body.

Everything known is the tathagatagarbha.

The Buddha said, “The tathagata-garbha is the cause of whatever is good or bad and is responsible for every form of existence everywhere.

It is like an actor who changes appearances in different settings but who lacks a self or what belongs to a self.

Because this is not understood, followers of other paths unwittingly imagine an agent responsible for the effects that arise from the threefold combination.

When it is impregnated by the habit-energy of beginningless fabrications, it is known as the repository consciousness and gives birth to fundamental ignorance along with seven kinds of consciousness.

It is like the ocean whose waves rise without cease.

But it transcends the misconception of impermanence or the conceit of a self and is essentially pure and clear.

The seven kinds of thoughts of the remaining forms of consciousness—the will, conceptual consciousness, and the others—rise and cease as the result of mistakenly projecting and grasping external appearances.

Because people are attached to the names and appearances of all kinds of shapes, they are unaware that such forms and characteristics are the perceptions of their own minds and that bliss or suffering do not lead to liberation.

As they become enveloped by names and appearances, their desires arise and create more desires, each becoming the cause or condition of the next.

Only if their senses stopped functioning, and , and they did not distinguish bliss or suffering, would they enter the Samadhi of Cessation of Sensation and Perception in the fourth dhyana heaven.

However, in their cultivation of the truths of liberation, they give rise to the concept of liberation and fail to transcend or transform what is called the repository consciousness of the tathagata-garbha.

And the seven kinds of consciousness never stop flowing.

And how so?

Because the different kinds of consciousness arise as a result of causes and conditions.

This is not the understanding of shravaka or pratyeka-buddha practitioners, as they do not realize there is no self that arises from grasping the individual or shared characteristics of the skandhas, dhatus, or ayatanas.

~Lankavatara Sutra

A buddha knows conditions as a buddhafield; that is the emptiness of conditions.

For a better easier non Buddhist perspective of such an entity search for "Boltzmann brain".

First, One mind does not mean number like one, two, three etc.

Second, emptiness and non-emptiness exist at the same time

OK let discuss one mind using the real life phenomena just to explain it. But I will not go too much into Buddhism, because the point is to clear your confusion of the term!

Example:

You start learning martial art. At first you have to learn all the basic moves. To defend which move you have to use, to attack which move you have to use. You practice these moves slowly, and the teacher sometimes acts like a dummies so you can demonstrate your work. When the teacher does this, you have to think which move you have to use. Your thinking and your move are not in sync!!!

But when you're good enough, and when you are in the real fight, your moves just comes out naturally without thinking! Of course, there is "the think" in your mind and there is "the move", but they happen so quickly, that you feel as if the move is happening without thinking! This is when you call your mind and your move is ONE. There is emptiness between the thinking and the move right now (there is no distinction between your thinking and your move they are merged into one, so the space or time between the thinking and the move is emptiness)

You see that there is one mind, it's also the emptiness! You see that one mind is non-emptiness, but there is emptiness of differentiation.

Do you see that the differentiation you have at first (which move to use) is the burden? Once you get rid of this differentiation you are now liberated?