An interesting question! This can be answered in two ways.
Answer #1: there is no conceptual gap
You note that the light clock argument is used to motivate a lot of results in special relativity, but dark matter doesn't interact with light. But the light clock argument is not really that important within special relativity, and neither is light itself. In the modern formulation, special relativity is actually about Lorentz symmetry, which relates space and time in a particular way. If a Lorentz symmetric theory contains massless particles (whether photons or otherwise), then you can prove on general grounds that they must travel at a universal speed. Also, you can deductively prove that all <PERSON> symmetric theories display time dilation, length contraction, and all the other familiar effects. The light clock is used in introductory courses simply because it's the easiest to understand. But moving pendulums tick slower too, as do moving mass-spring systems and LC circuits. These are all examples of the general symmetry at work.
Answer #2: there is a testable gap under exploration
If you were watching carefully, you'll note that I've just replaced one assumption with another. We can deduce that "dark matter clocks" behave as we'd expect in special relativity, provided that the theory of dark matter is <PERSON> symmetric. And how do we know that? We don't!
The vast majority of theories of dark matter simply assume <PERSON> symmetry. 562
That's not because we're sheep mindlessly reciting dogma, it's because we know too little about dark matter, so we need to make some concrete assumptions in our models, or else there will be far too many possibilities. Once you break <PERSON> symmetry, there are a vast number of ways it could be broken -- and many other symmetries you might as well break too. Of course, that doesn't mean you're not "allowed" to consider this possibility. For example, this recent paper considers precisely the scenario you are describing, and also proposes possible experimental signatures. But I wouldn't call this a solution to an "unsolved problem". It's more like one of many possible avenues of exploration.
I don't think many people study this kind of thing, because it seems too general and hard to test. It just seems a bit premature. (Wouldn't it be a lot easier if we discovered at least one concrete thing about dark matter first?) However, there have been many tests of potential Lorentz symmetry violation using the particles we already know about. Lorentz symmetry has been more stringently tested than almost any other symmetry in physics, and it has passed with flying colors every time. Of course, it could always potentially fail the next test, and then Lorentz violating dark matter would become a very interesting topic indeed. 484