Now, Rauser makes an error in explicating this and an error in objecting to it. Let's take the former first.
(1) Rauser says:
Let's begin by defining the concept of nature. A "nature" is merely a kind-essential set of properties. To exemplify a human nature (and thus to be human) means that you exemplify the set of properties essential for being human. The same goes for any other nature: tree nature, rock nature, divine nature, and so on.
This is an extremely anachronistic way of describing how nature is to be understood in this context. One will search in vain for any Greek philosopher with whom the Fathers might have been directly or indirectly familiar who thinks of nature (physis) as a set of properties. Precise definitions are not always forthcoming, but 'human nature' is what makes you human and in particular makes it possible for you to act in a human way. That is, it is a principle of action and passion, of operation and potential. This is actually quite important: properties as we usually understand them are things you have. They don't do anything. But repeatedly, over and over again, the Church Fathers speak of nature as something that does something. As Leo says, "Each form does the acts that pertain to it."
This is key for understanding where Rauser goes wrong in the objection.
(2) Rauser goes on to suggest the problem for the reduplicative view:
The problem is that it is not the natures that exemplify their constitutive properties, rather it is Jesus. It is not the human nature that is ignorant but rather Jesus. It is not the divine nature that is omniscient but rather Jesus.
Thus it makes no sense to say that Jesus was ignorant qua his humanity but omniscient qua his divinity. You're still saying that Jesus was simultaneously ignorant and omniscient which is a contradiction.
We should at once be suspicious of this sort of reasoning; if it were the case then reduplication would never under any circumstances be legitimate. We could not say, for instance, that Tom has responsibility for the actions of the police insofar as he is a mayor and does not have this responsibility insofar as he is a father. But, of course, this is a perfectly sensible thing to say about Tom: he does have the responsibility for actions of the police insofar as he is mayor, and he does not have that responsibility insofar as he is a father, because his being mayor is the reason he has the responsibility, and the responsibility is not a paternal responsibility.
Thus we need to be careful when dealing with reduplication. You can only know something given your particular abilities. This is obviously true; from the act to the power to act is a very sure kind of inference. But people can (and do) have different kinds of abilities. I can have experienced Devil's Tower insofar as I have the ability to see it and not insofar as I have the ability to climb it (assuming I have the ability to climb it!). And thus every property exemplified that involves some sort of action, activity, operation, reception, potential, etc., involves, so to speak, the ability to do or undergo that particular action or whatever. So, for instance, the finitude of my human knowledge is due to the fact that I have a finite human ability to know. Thus, while I have the property of being necessarily not omniscient, I do not have it simpliciter; I have it because I have only a particular kind of ability to know things. Thus the property I actually have is necessary non-omniscience insofar as this ability is considered.
But what if I were to have two distinct abilities to know? Then I could in a legitimate sense be both necessarily omniscient and necessarily non-omniscient because, again, I do not have either of these properties simply speaking, but only in a qualified way: I can have an ability to know that is necessarily omniscient and an ability to know that is necessarily non-omniscient.
This shows, incidentally, that it is simply false to assume that it is necessarily the subject that directly exemplifies properties rather than the nature. This is clear if the powers and abilities pertain to nature -- and I have never come across a reduplicationist who did not hold this. 'Omniscience' applies directly not to a subject (of any sort) but to an ability to know, either in itself or as exercised: an ability to know might be capable of omniscience. There are cases where this is quite obvious: I am finite in the sense that my power to effect things is finite, and this would be true even if I were infinite in some other way. If we want to talk only about properties exemplified by subjects, then I do not have the property 'non-omniscient'; I have the property 'non-omniscient given my particular ability to know things'. Only given the reduplication can I distinguish distinct properties; given the reduplication, it's not difficult at all to do so.
The objection is that there is a contradiction even if the reduplication is accepted. But we see by reflecting on this that this is entirely false. And indeed this is quite obvious: you can only get the contradiction by dropping the qualification added by the reduplication, in effect changing "non-omniscient given this particular ability to know" to "not omniscient given any abilities to know" and changing "omniscient given this particular ability to know" to "omniscient in at least one ability to know". This does indeed create a contradiction; but it is quite clearly not what the reduplicationist is saying.
Reduplication, of course, does not explain anything, or at least, it only explains why something is not contradictory if reduplication is allowed; it is purely logical in character. This is why the objection is problematic from the get-go: it is really an attack on reduplication as such, which is absurd, since reduplication in general admits of a perfectly respectable (if occasionally somewhat complicated) account. Someone who rejects the reduplicationist move should be attacking the reasons for reduplicating here in the first place, not arguing that reduplication still allows contradiction when it obviously doesn't. Since the reasons for reduplicating are laid out quite clearly in the conciliar letters to Cyril, the Tome of Leo, the Chalcedonian Definition, the Letter of Agatho, and the Definition of the Third Council of Constantinople, and each one of these documents itself already uses some reduplicative expressions at least on occasion, and even more often uses expressions that can easily be re-expressed reduplicatively, rejecting reduplication tout court means rejecting a massive amount of conciliar Christology. What Rauser is actually advocating is straightforward Monophysitism: his argument is easily shown to be a version of a standard Monophysite argument against Chalcedon (which they did indeed accuse of Nestorianism); it is unfortunate, after having made so much fuss about the Nestorianism of reduplicationism, that he did not think to mention the fact that treating the matter the way he does makes it impossible to repudiate Monophysitism. This is information that people might think important.
The chief problem with Rauser's post is that he presents the matter as if it were simple and easy, completely glossing over how much has to be given up if the objection is right. There are good, solid reasons why reduplication (in various forms) has been so popular a response through the ages, and it is these that need to be addressed.