There are no counterexamples to Kant's "argument" because it is not an argument. It is a view of predication under which being/existence is not a "real" predicate discussed in Transcendental Dialectic (Chapter III, Section 4):
"Anything we please can be made to serve as a logical predicate; the subject can even be predicated of itself; for logic abstracts from all content. But a determining predicate is a predicate which is added to the concept of the subject and enlarges it. [...] "Being" is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing... By whatever and by however many predicates we may think a thing - even if we completely determine it - we do not make the least addition to the thing when we further declare that this thing is. Otherwise it would not be exactly the same thing that exists, but something more than we had thought in the concept: and we could not, therefore, say that the exact object of my concept exists."
One can disagree with Kant's view, or one can even agree with it, but still choose to treat existence as a predicate for technical convenience. It is unclear what path you take in your example. Of course, we can think of listing "characteristics" (Kant's "logical predicates") of an object in a database, and one of those characteristics could be "exists in nature" (like horse) or "fictional" (like unicorn). What Kant is talking about is not such technical convenience that "abstracts from all content", but rather descriptive ("real") predicates that constitute our conceptions of objects. Be it horse or unicorn we do not learn anything new about our conception of them by discovering that such an animal occurs in nature. In fact, we need to form a conception first, and then go out into the nature to find out if an object of this conception exists there. Kant used this view to object to the ontological argument, which first declares existence to be part of the conception of "God", but then infers his real existence from it.
Kant's view was incorporated into the standard predicate logic by Frege and Russell, who treat existence not as a property of objects (predicate), but as a property of properties (quantifier). It applies to a list of properties ("conception") and returns true or false depending on whether there is an object in the domain of discourse which instantiates that list. Of course, existence can still be treated as a "characteristic" of objects with the help of identity, ∃x(x=a) expresses that a exists. But on a closer look what is being asserted is not that a exists but rather that the predicate is-a ("=a") is instantiated. There is no way to say that some a does not exist, ¬∃x(x=a) rather says that being-a is not a property of anything. One can object to such logicization of Kant's view (against his express words, apparently), but this is the prevailing interpretation of what he meant even among opponents of his view. Kant's notion of "conception" was developed into the descriptivist semantics by Russell, which came under heavy criticism, especially by Kripke in 1970s, see Kripke’s Attack on Descriptivism by Speaks, who offered an alternative theory of reference. Arguing with Kripke over Existence on Maverick Philosopher quotes him on existence specifically:
"To deny that it [existence] is a first-level concept is to deny that there is a meaningful existence predicate that can apply to objects or particulars. One cannot, according to Frege and Russell, say of an object that it exists or not because, so they argued, everything exists: how can one then divide up the objects in the world into those which exist and those which don't?"
An alternative approach to existence is pursued in logics of non-existent objects, which go back to Meinong with more recent versions developed by Parsons, Priest, and Zalta. There one does have the existence predicate, denoted E!, in addition to the existential quantifier ∃. Such logics distinguish between "there is" and "exists", where the former applies even to imaginary objects like unicorns and Pegasus, while the latter is something more real. In paticular, ∃x(¬E!(x)) expresses that there are objects that do not exist. But even these logics may not be in a real disagreement with Kant, but only regiment the use of "predicate" differently. This is subtle, see Berto's Existence as a Real Property: The Ontology of Meinongianism for a more radical view that traces Kantian view of existence all the way back to Parmenides:
"What the standard Platonic-Aristotelian interpretation puts in Parmenides’ mouth may be phrased by claiming that, for him, any concept including or entailing nonbeing in any form applies to nothing... much contemporary philosophy is dominated by what one may call the Parmenidean Thesis: everything exists... The philosophers we are about to meet have reinforced the Parmenidean position in two moves: (1) they have expanded the slogan “Existence is not a predicate” into the thesis that existence is (nothing but) property-instantiation; (2) they have also explained existence in terms of the logical notion of quantification. Both moves were made mainly by Frege and, derivatively, by Russell.
[...] Here comes a non-Parmenidean approach. To begin with, “exists” is a predicate of individuals just like the others – a predicate for real, not only from the point of view of our ordinary language’s surface grammar. It is a predicate in the same sense that “eats”, “flies”, and “is a man” are... The motto is Alexander’s famous one: “To be is to have causal powers”... The Parmenidean conception, once forced to admit that even from its viewpoint existence can be a property of individuals, still explains it away, defining it via the existential quantifier and identity, that is, via logical notions. But existence is not taken as a logical notion from now on: if to exist has to do with the enjoying of causal powers, whatever these actually are, they are not logical features."
