What is the meaning of "dhammā" in the context of the four satipaṭṭhāna?

Contrary to the assumption in question., it seems plainly obvious the Satipaṭṭhāna Sutta is not the Buddha's teaching. For example, how could the five hindrances be an object of the 4th satipatthana when the five hindrances must be dispensed with prior to the 1st satipatthana? Or how can the realisation or experience of the four noble truths pass away for an Awakened One?

Ajahn Buddhadasa suggested the Satipatthana Sutta is merely a list of disconnected dhammas. Ajahn Sujato has labelled the Satipatthana Sutta the 'Piltdown (Hoax) Sutta'. Surely, a Buddhist of reasonable discernment can see the Satipatthana Sutta is taught contrary to how the Buddha taught Dhamma, namely, it is taught in the wrong sequence pertaining to cause & effect.

If the Anapanasati Sutta is used as a guide, the meaning of dhammānupassana is clear. "Dhamma" refers to the Supramundane Realisation & Fruit declared by the Buddha.

I'd go with "the contents of mind", as opposed to the overall condition of mind (the point of the third "foundation", as is evident from the elaboration: attraction, aversion, doubt, agitation, sloth etc.).

You seem to assume that the first three foundations are all ultimately objects of mind, but I don't think that's a perspective taken here.

The matrika seems to be a simple progression from coarse phenomena to more refined ones. First, there is the body with its sensations (vedana), then there is the mind, with its contents (dhamma).

The word dhamma literally means "that which is held". In phenomenological context, it means "that which is held by the mind". So in case of satipatthana, dhamma means "objects of mind", or "contents of mind".

Uncommon Wisdom: Life and Teachings of Ajaan Paññāvaddho

"Wisdom

The path of wisdom practice—investigating inward from the coarsest objects to the most refined—is exemplified in the Four Foundations of Mindfulness or the Four Satipaṭṭhāna: body, feeling, citta and dhammā. Here we have a path leading progressively from the external to the internal. The body, being external, is the most obvious. Going inward, feeling is represented by the more subtle feeling-body.

More subtle still is citta or states of mind. Lastly, there is dhammā, which is the content of the mind—the subtlest phenomena of all. Each of these four is a domain of personal experience, and each is a mode of establishing mindfulness...

... Various factors make up the processes of the mind. There are feelings and memories; then the two major factors that make up thought, which are sankhāras and dhammas. By sankhāras, we mean the mental formations that create thoughts and ideas. When sankhāras group together, they form states of mind, which are combinations of many different factors, like anxiety, anger, conceit, compassion, concentration and so on. In the case of establishing mindfulness in the domain of mental states, we see the arising and ceasing of the factors that comprise those states, and the relationship between those states and our experience of body and feeling.

The fourth domain of satipaṭṭhāna, dhammā, refers to the content of the mind. The dhammas are the basic elements that make up mental formations and states of mind, and those elementary factors cannot be reduced any further. They are qualities and faculties that arise in the mind. For instance, pure hatred and pure greed are dhammas. They simply arise on their own, and they cannot be dissected any further....

...When mindfulness is well established in the internal body, the relationship between feelings and the states of mind that define and interpret them becomes apparent. In other words, the way we interpret the feelings that define how we experience the body is determined by our mental state. From that understanding, we realize that the mind is the true basis of feeling. As our contemplation moves deeper into mental states, our attachment to the domain of feeling—an essential aspect of our personal identity—starts to fade into the back-ground. Feelings now appear external, and the primary focus turns inward to our mental states.

With the establishment of mindfulness firmly based in the domain of mental states, the subtle phenomena that make up the content of the mind are more readily perceived. These mental phenomena are far more refined than the states of mind they bring into being, and therefore more “internal” in relation to mental processes. In the final analysis, attachment to these subtle phenomena must be overcome in order to attain the mind’s liberation..."

From my perspective this is a very neat explanation. I include the following as supplementary for those not so familiar with their teachings.

From the tornado of self. https://www.youtube.com/watch?v=9ldqXnI6mRo

“The Citta as referenced in the forest tradition as the structure to which mind finds support. Citta is nibhana in the state of the complete removal of sankhāras. When the citta is ruled by the fundamental ignorance of avijjã and tainted by kilesas, it pushes the khandhas to do unwholesome actions of body, speech and mind. When Dhamma is in control, wholesome actions occur.”

From Uncommon wisdom.

“By the same reasoning, the Noble Eightfold Path is not a path that one travels along as one would a road or a walkway. Rather, the Path is set up as a mode of transcendence. When we have done the work to set the Path up correctly, it acts like a channel for transcendent states of mind to arise—Sotāpanna, Sakadāgāmī, Anāgāmī and Arahant. Because of that, all the path factors arise simultaneously. It is a difficult feat to accomplish because we must get all of those factors just right at the same moment. Having done the work, when the right conditions arise, they will all come together and bring forth the path moment.

In order to accomplish this, we must gradually develop all of the conditions which are necessary for that moment to take place. It involves not only formal meditation practice but all of our activities throughout the day. Effort and wisdom must be present at all times in order to turn every situation into Dhamma.”

Does anyone have a proper explanation as to why the fourth set is called "dhammas"?

Because the 4th category is not about dhamma (mental qualities or phenomena), it's ☸Dhamma-[teaching that leads directly to nirvana].

http://notesonthedhamma.blogspot.com/2019/07/proof-dharma-is-buddhas-teaching-not.html

The standard Theravada interpretation of the cryptic formula in 4sp satipatthana, is that one focuses on one of the frames to the exclusion of the other 3. But that's not the real purpose. The real meaning, is that one should see "body as body [as it actually is]". http://notesonthedhamma.blogspot.com/2019/05/the-real-meaning-of-cryptic-formula-in.html

For ☸Dhamma, the meaning of seeing ☸Dhamma as ☸Dhamma as it actually is, is very similar in meaning to the meaning of ☸Dhamma-anu-dhamma☸📐 {the practice of} dhamma/Dhamma that accords with {the Buddha's} ☸Dhamma. http://lucid24.org/tped/d/dhamma/index.html

EA, the MN 10 (satipatthana sutta) parallel in Chinese Agama, supports my interpretation Or at the very least, has an idea much closer to mine, than to Theravada. It looks like they're interpretation is that you take any dhamma [thing, quality, phenomena], and put it through a thorough ☸Dhamma investigation, after attaining 4 jhanas via 7sb. (chinese and english translation of passage here): https://notesonthedhamma.blogspot.com/2019/07/what-is-dhamma-category-in-4sp.html

my response to one of the comments below: in DN 22 though, the 4 noble truths and noble eightfold path description under Dhammaanupassana are just cut and paste jobs from other fundamental suttas defining what those factors are. Unlike the EA parallel where Dhamma anupassana is described where you're activing USING a Dhamma-teaching framework to analyze the phenomena. DN 22 maybe implies that, but the Theravada model is so massive and convoluted you can't tell for sure. In EA, for sure that's what they're doing. The problem is if you take DN 22 as viewing dhamma-phenomena viewed through Dhamma-teaching framework, then how do you approach the other 3 satipatthana where that wouldn't work? It doesn't say kaya-Dhamma-anupassi, vedana-dhamma-anupassa, etc...