The best argument against apathetic agnosticism is Pascal's wager. The cost of a mistake is too high; if there is even a minute, nonzero probability that there is a real non-apathetic supreme being, then it is extremely important to know where you stand with him/her/it. (Note: this conclusion follows from the wager regardless of whether we compare two possibilities or myriad, as long as one of them is "there is a God who actually cares what you do/believe".)
Pascal's wager does not work to refute agnosticism or atheism generally. However, apathy regarding the question is off the table. To all of you who take offense at Pascal's wager: Please take a deep breath and notice that you are not actually apathetic. You do actively care whether or not a religion like Christianity or Islam is true.
So apathetic agnosticism makes a very strong positive assertion: there is no non-apathetic deity. If you thought there was a small chance that there was such a deity, you would not be apathetic about the question. You would take the time to evaluate the evidence thoroughly.
Therefore, apathetic agnosticism is not actually apathetic; it is masquerading as something that it is not. Articles one and two are both just as strongly dogmatic as the religious worldviews that apathetic agnosticism purports to be a non-dogmatic alternative to. It is only apathetic regarding a question of the existence of univolved and impersonal supernatural powers, which is a question which I doubt anyone considers is of much importance.
There are other problems with each article:
Article 1: The existence of a Supreme Being is unknown and unknowable. This implicitly assumes an evidentialist epistemology, which has its own host of problems. Most obviously, it is self-consuming--what evidence do you have to conclude that you should only believe in things you have evidence for? It is also not, strictly speaking, true that there is no evidence for God. Hundreds of books have been written on the topic and I shan't rehash it all here. One might reasonably say there is no good evidence for a God, but that's a statement which requires a non-apathetic evaluation of the purported evidence.
Article 2: If there is a Supreme Being, then that being appears to act as if apathetic to events in our universe. This one is really really questionable. Let me break down the commentary:
All events in our Universe, including its beginning, can be explained with or without the existence of a Supreme Being.
One might truncate this to "All events in our Universe, including its beginning, can be explained" and we already have a questionable, probably false statement. How can one justify the assertion that all things in our universe are explainable? Who believes that? Given that you haven't observed all events in our universe, especially its beginning, it is a violation of your principle that "we can only state with assurance that we do not know." to state that you know all those events you've not observed are equally explainable in both theistic and non-theistic terms.
Furthermore, this is simply a false statement about the origin of the universe. The big advantage of theism is that it does have an explanation for the origin of the universe, while atheism does not. The universe obviously must either have always existed, come into existence for no reason, or originated from something that is not "the universe". In the first two cases, its origin does not have an explanation. In the third, the explanation starts to sound awfully theistic. Some more lucid atheists have recognized this, e.g. Steven Hawking once said "If you like, you can call the laws of science 'God', but it wouldn't be a personal God that you would meet and put questions to." Perhaps the apathetic agnostic meant to say, "All events in our Universe, including its beginning, can be explained with or without the existence of a non-apathetic Supreme Being." But in that case, since you are asserting that the origin of the universe is explainable, you are essentially forced to assert the existence of a deity, though perhaps of the non-personal type.
Thus, if there is indeed a God, then that god has had no more impact than no god at all.
I guess I don't see how this follows from the first sentence. For example, if there is an afterlife, then we wouldn't necessarily expect some measurable difference in this world, but God may have a great deal of impact on your post-death experience. So even taking the first sentence about explainability for granted, this conclusion only applies if we also make some assumptions like materialism.
To all appearances, any purported Supreme Being is indifferent to our Universe and to its inhabitants.
Or maybe he's happy with the way things are playing out? Or perhaps, he's waiting for something? To say that the evidence is consistent with this conclusion cannot be used to rule out alternative hypotheses. It's also a misconception of religious theism to suppose that the primary purpose of believing in God is the explanatory power of that belief. Among Christians, Jews, Hindus, Buddhists, and Muslims that I have met, I don't think a single one would say that their reason for their faith is its ability to explain natural phenomena.
Article 3: We are apathetic to the existence or nonexistence of a Supreme Being. If the first two articles were reasonable, this would be reasonable too. But, Pascal's wager shows that it's only reasonable to be apathetic about the existence/nonexistence of an indifferent Supreme Being. It is not reasonable to be apathetic about the existence of a God who cares. An apathetic agnostic is not apathetic regarding that question.