Textual variants presuppose texts. If one thinks the way in which people know, in an internalist and infallibilist sense, what is and is not the content of special divine revelation is through textual criticism - through identifying textual variants and selecting which of them is "authentic" to the "original" text - he is not beginning with special revelation, he is coming to a conclusion about it. Of course, a conclusion is only as good as its premises, which leads to consideration of one's ultimate or foundational premises. Are they capable of yielding a satisfying philosophy?
I've argued elsewhere on this blog and so will not now attempt to reproduce why I think special divine revelation - the extant extent of which is concretely codified in the Scriptures - is the premise with which one must begin in order to intentionally defend his beliefs with full assurance. This is not to say that my arguments are reasons for this foundational belief, but as Gordon Clark put it, by the systems they produce, axioms must be judged. If an axiom doesn't produce a sufficiently coherent system, can't account for certain worldview necessities, isn't sufficiently explanatory - however you want to phrase it - the axiom itself fails to give us knowledge of the infallibilist and internalist variety. Of course, one may object that we need any such knowledge, which is itself another discussion, one about needs. One can also be mistaken in his judgment, and the "tests" - for the present lack of a better term - do not imply that the falsifiability of an axiom is a live possibility. Yet there is practical use for these tests, and apologetists ought to be in the business of being practical whenever possible.
If, then, a man can't infallibly or with full assurance defend how he knows anything - including the idea that there are texts or textual variants in codifications of special divine revelation - apart from ultimately presupposing a special revelation, then I don't see that textual criticism presents an epistemic problem. Textual variants only pose an epistemic problem if one assumes that in order to identify, recognize, know, or, in particular, defend special divine revelation, one must infer the content of it from a group of texts which sometimes do not conform. In that case, you're attempting to reason to what special divine revelation, not reason from what special divine revelation is. I still agree with these aspects of my post from a few years ago.
However, I also think this is a subject for which the externalism-internalism distinction is relevant, and that requires an update to my former position of occasionalism (link). What we know are propositions. Externalism is the theory that we can, to varying degrees and depending on the justificatory factors involved, know or be epistemically justified in our beliefs due to something to which we don't have cognitive or reflective access - say, a causal process. We can think about or reflect on a causal process, but we can't re-experience it, whereas we can periodically access or experience the same beliefs. A causal process might be considered able to epistemically justify us because that process in general produces true beliefs in the mind of the person who undergoes it. The causal process tracks truth, whether we are aware of it or not.
That kind of "epistemic justification" allows for the possibility of our knowing what are generally considered "common sense" beliefs. I'm typing on my computer, you're reading a blog post, etc. The causal process by which we know these propositions is usually physical media. But the chain of causes which produce a belief need not be evidentiary reasons for my belief. For example, while God is the ultimate cause of all things, not all people's beliefs will be reasoned from or evidenced by a belief they may have - or, more pertinent to this example, may lack - about God. Similarly, while I may have a sense experience which causes a belief in divine revelation, I needn't infer my belief in divine revelation from a belief about my senses. So if, after a causal process consisting of the examination of textual variants, you believe something to have been divinely revealed, that doesn't require you to epistemically ground your belief regarding the content of divine revelation on a belief about that causal process. Again, I would argue a belief about that or any causal process is itself infallibly defensible only by ultimately appealing to special divine revelation.
That doesn't mean the causal process is irrelevant to your belief. If we have a belief that certain causal processes track truth better than others, it makes sense to position ourselves and those around us to more often experience the better kinds of causal process. If I want you to know about the Grand Canyon, I may talk to you about it or show you a picture of it, but I wouldn't shut your eyes or close your ears while I did those things. I think sense experiences often cause true beliefs. If I want you to know a truth, and if I believe there is a kind of experience which may be useful in producing a true belief, I'll do what I can to help you experience that.
I believe the above illustration provides a fair analogy of how I think we can regard at least one goal of textual criticism. There are textual variants among what copies of Scripture we have. Some do not affect the meaning of a passage. Some are evidently the result of mistranslation. Some are more significant in implication - the variants may affect the meaning of a passage, or they may exhibit disagreement with other texts about whether a passage is even canonical. Thus, while I think the goal of the textual critic shouldn't be to collect texts, compare and contrast them, and use that as an evidentiary basis to infer or reason to what has been specially divinely revealed, there certainly would be use in disposing ourselves and others to a causal process which tracks truth about what has been specially divinely revealed and codified in physical media - in this case, texts. So one function of textual criticism could lie in its capability to cause externalist knowledge of special divine revelation. In any case, there is certainly some apologetic role textual criticism may play within one's worldview, so long as it is remembered that apologetics is subservient to and in fact derives from one's epistemology (link).