Do we have an inbuilt spiritual sense as part of our consciousness that there has always been more to our perceived reality than we can prove?

I will only address the first question that seems to me to be implicit in the statement that the concept of Dao seems to be an example of a "trans-empirical" reality. I will also assume that by "trans-empirical reality" something like Kant's noumena is meant, so not some kind of supernatural reality existing separately from empirical reality.

The daoist concept of the Way is similar to limit concepts, very general boundary concepts, such as the concept of determinate, single "things", or "reality", the concept of "the world is everything that is the case", concepts that seem to be implied or presupposed in all forms of perception or thought. Are those concepts "trans-empirical"? Yes and no. Yes, in sofar as any empirical observation also presupposes them. (In psychological terms: before we can be aware of anything, we need to have some object constancy.) But as such those concepts are also empty abstractions. There are things. We perceive things. But only if we already, in some environment, for some mode of perception, have a concept of determinate things. Which just means: our perceptions have structure. So, no, in sofar as those concepts still refer to empirical reality.

The Dao is more than just a boundary concept, however. It refers to nature seen as a dynamic, self-organizing process. It is not "something" outside of this process, it's not something deeper or higher, something apart or separate from what happens in nature. It's sometimes spoken of as something separate, but all of those kind of descriptions are immediately relativized, taken back as inssuficient. The core statement is:

道法自然
The Way follows what-is-so-by-itself

The word 自然 zìrán literally means "what is so by itself"; it also means "nature", as what is spontaneous, natural. This idea of natural order is very much inspired by biological processes, patterns of mutual dependencies, seeking to maintain or restore homeostatic balance, rather than by a mechanistic cause and effect.

One of the most remarkable aspects of the early daoist idea of nature is that it rather explicitly denies that processes in nature have any (inherent or externally-imposed) goal or purpose. This breaks with what may be an innate tendency in humans, namely, to explain events in teleological terms^A, or even stronger: in teleological anthropocentric terms. A rather clear expression of this is Laozi 5:

天地不仁，以萬物為芻狗
Heaven and earth art not humane,
they treat the ten thousand things like straw dogs.

"Straw dogs" is thought to refer to dogs made of straw that were used in certain religious sacrifices (replacing actually living creatures as sacrifice), and that would be discarded as rubbish after the ceremony^B.

The spiritual dimension here, in my opinion, is simply that it sees human beings as part of nature. Nature can destroy us, of course, but that doesn't mean it's the "enemy" or something that needs to be tamed or subjugated. Ultimately, we too are compostable. That's just the way it is. And there is nothing wrong with it: it doesn't need to detract from any values we may experience.

(A) See for instance: Debohra Keleman, The scope of teleological thinking in preschool children (1999). A funny (or not so funny) example of teleological thinking in very early Chinese thought can be found in the oracle bone inscriptions. One oracle asks, for instance, whether or not the King's tooth ache is due to the King having angered ancestor So-and-so.

(B) Wikipedia has an interesting lemma on "Wang Bi, who wrote an influential early commentary on the book of Lazozi (3d century CE), explains the lines as:

Heaven and Earth do not produce grass for the benefit of cattle, but the cattle [still] eat grass. They do not produce dogs for the benefit of men, but men [still] eat dogs.

This clearly underlines the break with teleological explanation.

1. “The theory of trans-empirical reality” is what in European philosophy is named metaphysics. It is a characteristic of metaphysical entities in the sense above that they cannot be registered by our senses.

2. If you read the Tao Te Ching you find the first line:

   “The Tao that can be told is not the eternal Tao.”

   So it is difficult to make words about a subject which refutes any talking about. In particular people cannot exchange their findings in a rational way.

   Such concepts like "Tao" in China or "Brahman" in India are always at risk that each reader creates their own interpretation and starts to develop their own ideas in a cloudy domain.

   I consider a good commentary book “Trey Smith and Alex Paul: Around the Tao in 80 Days".

3. I do not see any biological basis that humans are born with an inbuilt spiritual sense. But humans are able to make extraordinary experiences and interpret them in different ways.

Humans as a species possess three qualities that are relevant to your enquiry- an ability to imagine and speculate, a tendency to be impressionable, and an imperfect ability to reason logically. The first two of those qualities can be viewed straightforwardly as advantages from an evolutionary perspective, while perhaps the third might be considered unfinished business in that regard. How those qualities manifest differs from person to person, culture to culture, time to time, and so on. In order to answer your question with a definitive yes, you would need evidence to show that the tendency among some people to speculate on spiritual matters is not just a particular manifestation of the three more general qualities I mentioned. Given that many people profess not to be in the least spiritual, I suspect the proof you seek might not be readily available.

Re. "trans-empirical reality" — Certainly there are things that we do not know about. From an individual perspective new things emerge from a place where things are not known, where nothing is known, and which appears as nothing [wu] because nothing of it is known.

Making sense of what appears from nothing [wu] has to do with Dao. For instance, in Heidegger's Hidden Sources chapter 4 'Dao: way and saying' (page 40) Reinhard May quotes Martin Buber from the afterword of Buber's edition of the Zhuangzi:

The word dao means way, path: but since it also has the meaning of speech [Rede], it has sometimes been rendered by ‘Logos’. In Laozi and his disciples, where it has always been developed metaphorically, it is associated with the first of these meanings. And yet its linguistic atmosphere is actually related to that of the Heraclitean Logos (100).
Dao does not mean any kind of world-explanation, but rather that the entire meaning of Being [Sinn des Seins!] rests in the unity of true life, is experienced only in that unity, and that it is precisely this unity taken as the Absolute. If one wants to look away from the unity of true life and contemplate what ‘underlies’ it, there is nothing left but the unknowable, of which no more can be said than that it is unknowable (101).

So there, Dao is linked to the way and being. Being is often linked to nothing, but generally in the sense of no-thing. In Hidden Sources chapter 3 'Nothing, emptiness, and the clearing' (page 26) we see wu used in this way.

Chapter 40 of the Laozi goes on to elaborate the idea that everything [alles Seiende] in the world, or all things [wan wu] originate [sheng] from being [yu], and that being [yu] originates [sheng] from nothing [wu]. Von Strauss translates: ‘All beings originate from being, / Being originates from non-being’.

Also in the Chuang Tzu Book XXII ¶ 10

Was that which was produced before Heaven and Earth a thing? That which made things and gave to each its character was not itself a thing.

There is a metatheory, called "The Perennial Philosophy" by Huxley, in his book of the same name, which holds that there are spiritual realities, deeper than ordinary reality, and that these have been accessed, repeatedly in history, and across societies, cultures and eras. Examples of philosophies counted as the fruit of such insights include Platonism and Neoplatonism, Taoism, Sufism, Zen Buddhism, Kabbalah and mystical Christianity.

If you ascribe to this, it would suggest that there such a spiritual sense does exist, but that it might not be equally accessible by all people at all times.

Do we have an inbuilt spiritual sense as part of our consciousness that there has always been more to our perceived reality than we can prove?

Perceived Reality is a very big thing. See as part of some experiment i spent some nights alone in cemetery and the things I experienced there add a TOTALLY NEW DIMENSION TO "PERCEIVED" AND "REALITY". Now Ghosts and God are no longer myth. See as far as proving is concerned, "DEFINING" is the biggest problem. Tell me what is Anger, Define it? Don't tell me the effects, tell me ANGER. Although EVERYONE Experiences it, no one can Define, let alone prove.

Now as far as "Spiritual" is concerned, your Spirit, your Soul is SOURCE of all these that you feel. To get there, to understand it, to have some conception of it, you can't proceed by looking at effects. You'll have to look at it in it's nascent still state. Once you see that, you'll know there is no way to say much about it. And IT'S SUBTLE, VERY VERY SUBTLE. Your Soul Works through you and you never notice. BUT WE DO HAVE A LINGERING SUSPICION THAT THERE IS MORE TO ALL THIS. This in a way is how that "Spiritual Sense", we do have some idea but despite looking at it dead on, we failed to notice it, the Soul, the "Spiritual" part. Also some of Soul's wisdom seeps through that's known as intuition, sense. SPIRITUAL SENSE.

You can see and Experience Soul, it takes time but possible.

Yes there is a Tao, but it cannot be spoken of, for the spoken is not the Tao...