The problem as I understand it.
This is a question is ultimately about substantiation.
In using the term unique particular, you seem to be referring to the notion that there may be some sort of irreducible particle, a particle that would give objects and phenomena (dharmas) true substance, or something that would be conceptually equivalent to a true ... say "atom" that allows us to build a "true" chair. The issue at hand being, of course, that since objects all are found to be empty upon final examination, that we still have to contend with where apparent "chariness" comes from, and therefore also all such-ness, "that"-ness, and ultimately the phenomenological experience of the apparent self. The existence of such a particular would have far reaching implications and consequences.
This style of inquiry is a search for svabhava, true self-substance. It is looking for the thing that exists from it's own side, a truly independent phenomena, the source of substance in the world. It is equivalent to the search for a substantive true self, and to the search for the god particle. It's hunting the Unicorn. Since we cannot find such a thing in the macroscopic reality, or simply the everyday world of objects and events, the notion that there is a fundamental, unique, and "bare" particular underneath it all, acting as the building block, would seem to solve the dilemma of the inherent uneasiness found in coming to terms with an "appearance only" based reality, particularly for the realists. Here is where the Buddhist "atomists" come from, and consequentially all fundamental materialists.
The logical conclusion of that solution to the problem...
The problem is that such a unique, or bare particular would still be an individual in possession of the characteristics of bareness, uniqueness, and particular-ness as its intrinsic nature, and to find the true individual behind the characteristics, or to find the individual holding that nature, it would have to be stripped back down beyond said qualities that it therefore has, requiring another bare/unique particular behind it to give it substance. We have to follow this chain of inquiry back infinity, never finding a termination point.
This issue is very skillfully dissected in Jan Westerhoff's Nāgārjuna‘s Madhyamaka: A Philosophical Introduction, published by Oxford Press.
"...if it is a bare particular by svabhāva and being a bare particular is therefore its intrinsic nature ... we can ask what the individual is in which being a bare particular inheres, and then we are well on our way to an infinite regress... we have to assume that the individual has some determinate nature due to which it is a bearer of its properties and the difficulty will just reappear with whatever we take such a nature to be."
So, if we have an individual, and that individual has the property of being a bare, or unique particular, then without that property what ground does that individual stand on? It's the same as saying that a chair cannot exist without its characteristics, and the characteristics do not exist as such without the chair. This is, from a Madhamaka point of view, the fundamental meaning of co-dependent origination, so from such a perspective if such a thing were to be found, that that would stand against the core Buddhist doctrine of sunyata.
The bare particular is landed on when examining an absolute creator, an absolute substance, a true, objective reality, the existence of real time, a true (meaning complete and consistent) axiomatic system, etc...
In the specific case of the bare particular it is shown that such a conceptualization is absurd, since we can conceive of a bare particular, but it falls into a co-dependent relationship with itself via its characteristics, and cannot exist outside of, or independent from mental conceptualization. In other words, it is mind dependent, and because of this, and the fact that this is essentially the only way to get at substance, so is every other level of such-ness, regards to the chair, or the self. You see? Without mind, without conceptualization inferring the existence of substance, of chariness, of self, there is no substance to be found, anywhere, with anything, beyond emptiness via dependent origination.
This is vast emptiness. This is the display of the true power of the mind and delusion. Somewhere, it is building the house, and the house has as its nature impermanence and suffering. The logical consequence, and final legitimate course of action is practice. So theory and practice do inform one another.
Such a particular would break causality, as a true existent cannot be altered from its nature, and cannot therefore be interacted with. IN order for something to be causally potent, it has to be empty.
Such a particular would also mean that there is inherency in life, absolute positions, a central way and meaning of life, and therefore true hierarchies of values in terms of absolute right and wrong, etc. It would, as an eventual consequence of this, justify all atrocity.
Also, the atom is a great example because it breaks down into these sub-atomic particles that blink in and out of existence, exiting as wave/particle "things"
with only probable substance, and no absolute essence beyond, or apart from, the probable.
Since empirically adequate simply means that a statement is such "if and only if everything that it says about observable entities is true", we have to ask, "what makes a true entity?" If we are going to ask that, we fall back on looking for the bare particular. In co-dependent origination we cannot find any true entities, so it becomes non-codependent non-origination from a certain light. So there is nothing "true" in an absolute sense to observe, just samsara. Because of this, every so called empirically adequate statement we could make about the observable universe would be a relative, conventional truth only. So, no. Empirically adequate does not, indeed cannot equate to unique particular, since it is a referent to conventional reality only, and therefore dependent.
The result of this is that we can regard all dharmas as dreams, and people have spent a great deal of their practice hours studying this approach. It is, as I understand this, the heart of such things as the Diamond Cutter.
TL;DR An empirically adequate statement can only refer to conventional reality, and as such, is a convectional truth and conventional designation itself, dependent on phenomena for existence, and therefore no... it is not a bare particular, or "true existent", or unique particular, or absolute truth as you posit. Such a thing cannot exist to begin with.This is a core fundamental of Buddhist thought, at least beginning with folks like Nagarjuna. We could assume all dharma's to be, in this way, empirically adequate, ultimately empty themselves. This is what makes Buddhist thought so sublime, and keeps the system legitimate. It is, in my understanding, the philosophical perfection in the system, and an important component of what makes it rise above "mere philosophy."