(N.b. there appears to be a similarity between the conventional/ultimate distinction you've explicitly mentioned and the Aristotelian-Thomistic distinction between accidental form and substantial form. In the former case, we may speak of artifacts like chairs, computers and clay pots whose telos is extrinsic and observer relative (to borrow a Searlian expression) whereas in the latter case we may speak of human beings and oak trees whose telos is intrinsic. That a suitably shaped piece of fired clay it taken to be a clay pot is thus a matter of convention, but that a particular man is a human being is not. The atomist is therefore someone who takes everything to be an accidental whole arranged out of whatever he takes to be atoms.)
"The Buddhist position presupposes that we can know that there are atoms, but claims that the composites we take to be made up of atoms are unreal.
"If they are unreal, then of course we cannot be said to know of them through perception. But the atoms are not perceptible either. So how could the Buddhist know of them any more than he knows the purportedly unreal composites? In fact it is only through the composites that we can know the atoms that make them up. Hence the composites must be real."
The Buddhist argument doesn't say that the composite isn't perceivable or doesn't exist, just that it only exists conditionally. Nobody denies you can't see the composite.
It's important in these conversations that the Buddhist position is always one of soteriological convenience. The point of conceptually separating conventional and ultimate existence is to point out that your mind and identity exist only conditionally. The suffering you experience is part of this conditionally existing group of mental processes. So therefore the causes for your suffering can be removed and that will remove the suffering which is conditioned on those causes.
I don't think anyone makes exactly this claim. From the article, which I presume draws faithfully from the source text[0]:
"Both sides make reference to ordinary objects, such as a clay pot. Both sides agree that the pot is ultimately made up of unobservable atoms. But the Buddhist reductionist says that those atoms are really all that exist, that there is no such thing as a composite whole over and above them."
The point of disagreement between the Buddhist reductionist and the Nyāya commentator, then, is over the existence of composites as bona fide things versus, say, perceptual illusions or whatever. Because both the Buddhist reductionist and the Nyāya commentator refer to composites, clearly both accept composites as objects of perception, so what is perceived is not the point of disagreement, in which case where does the misunderstanding over the conditionality of composites enter the picture? Again, the Nyāya commentator doesn't deny that composites are composed of atoms, nor that their existence is contingent or temporary. Additionally, it seems that if atoms are what ultimately exist, then there would appear to be at least some things which are not conditioned, if I understand the term.
Now, given that, the counterargument is epistemological in nature, specifically, that if what we know about the world begins with perception, and what is perceived are composites and not atoms, and that our knowledge of atoms is inferred from our knowledge of what is perceived, then it is incoherent to also deny the existence of composites because our knowledge of atoms depends on our knowledge of composites. To deny the reality of the latter is to deny the knowledge of anything that depends on the former being the case.
[0] https://www.amazon.com/Nyaya-sutra-Selections-Early-Commenta...