Is there an acknowledged unsolved problem in some cosmological theories with respect to "dark" matter and general relativity? Some mainstream cosmological hypotheses hold both that:

*

*general relativity is correct and universal; and

*a form of "dark" matter exists that is, in somewhat of a misnomer, "non-baryonic".

Let's be clear than I'm not asking about other cosmological hypotheses that seek to vary one or both of these.
The statement that the speed of light in a vacuum is observed the same in all free-falling frames of reference is axiomatic in general relativity.
Similarly, widely-used thought experiments in relativity include "clocks" made from mirrors bouncing photons back and forth and beams of light in the interiors of rocketships.
The "non-baryonic" matter is a bit of a misnomer because it is not quite baryons that this particular type of "dark" matter is postulated as not interacting with; nor is it necessarily postulated as being like (say) an electron (which is non-baryonic but not "dark").
It does not interact with photons, possibly moreover with few or even none of the Standard Model's gauge bosons at all.
(There are variants, but not interacting with photons is the common basis for the "dark" appellation.)
The apparent gap between these two postulates is that it does not seem possible for a "dark" matter system to have (some basic equivalent to) bouncing photon clocks or rocketships with light beam spots on their walls, and it does not seem possible for a "dark" matter system to observe the speed of light in a vacuum.
It would actually have to interact with photons to observe them; and it is axiomatic that it does not.
Have theoreticians acknowledged this as an unsolved problem in these cosmological hypotheses?
Lists of unsolved problems that I have found do not seem to touch on making these two postulates join together as a problem to be solved.
(Yes, the very existence of "dark" matter as a whole is an unsolved problem, but remember that I'm asking about proposed cosmologies that do postulate its existence with specific properties.)
I'm not looking for personal theories.
I'm looking for what the state of the art actually is in theoretical physics.
I'm looking for answers such as (possibly combinations of):

*

*Yes, they have acknowledged the gap and proposed a mechanism in the cosmology.
In which case: Who has? What hypothesis have they proposed?
A good answer would point to some papers and name names.
Answers such as "Theoretical physicist X has proposed in paper Y gravity as an indirect coupling/photon-like 'dark' bosons whose speed is also constant/et cetera." fit here.

*Yes, they have acknowledged the gap but no-one has yet proposed something.
A good answer would point to a paper listing this as an unsolved problem, specifically in these cosmologies that postulate this "dark" matter (not merely that "dark" matter as a whole is an unsolved problem).

*Yes, they have acknowledged the gap and this is actually considered a reason to reject the cosmology and substitute an alternative.
A good answer would again point to some papers and name names.
Answers such as "Theoretical physicist X has identified this in paper Y and rejected the existence of such 'dark' matter/proposed that general relativity is not universal and 'dark' matter is not subject to it/proposed that such matter is 'dim' rather than strictly 'dark' and could measure the speed of light." fit here.
It's not relevant whether that rejection is verifiable or the details of the alternative hypothesis; the point of interest being whether an explicit proposal to outright reject cosmologies on this ground, after identifying that it is a ground, even exists.

*No, they have not acknowledged the gap.
But they haven't argued that it isn't there.
This is basically uncharted territory, full stop.

*No, they argue that there isn't a gap.
(Included for completeness, but I think this answer to be unlikely.  It doesn't seem sensible to just declare by fiat that experimental results for normal matter hold for "dark" matter when one is defining "dark" matter in such a way that such experiments are impossible; and it's the experimental results failing to find different values of c that support the axioms of special relativity, and hence general relativity.)

 A: 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 Lorentz 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 Lorentz symmetric. And how do we know that? We don't!
The vast majority of theories of dark matter simply assume Lorentz symmetry. 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 Lorentz 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.
A: 
The statement that the speed of light in a vacuum is observed the same in all free-falling frames of reference is axiomatic in general relativity.

Einstein took that as a postulate, but you could also develop the theory in other ways. You can have a Lorentz-symmetric physics in which only sublight speeds exist; you don't need anything to move at exactly $c$ to have the same geometric spacetime structure.
Also, dark matter isn't expected to be totally noninteracting with ordinary matter, just very weakly interacting. Neutrinos are dark matter.
Also, gravitational waves propagate at $c$, and everything interacts with gravity. "The speed of light" is just a name; special and general relativity don't need a U(1) gauge theory to exist.

The "non-baryonic" matter is a bit of a misnomer because it is not quite baryons that this particular type of "dark" matter is postulated as not interacting with; nor is it necessarily postulated as being like (say) an electron (which is non-baryonic but not "dark").

"Non-baryonic dark matter" means dark matter that isn't baryonic, i.e. that isn't just hunks of rock that don't glow enough to be seen. "Non-baryonic" doesn't mean it doesn't interact with baryonic matter. It mostly doesn't, but neither do the rocks.
The category of baryonic matter is defined to include electrons, even though electrons aren't baryons. The terminology could use some work.

The apparent gap between these two postulates is that it does not seem possible for a "dark" matter system to have (some basic equivalent to) bouncing photon clocks or rocketships with light beam spots on their walls, and it does not seem possible for a "dark" matter system to observe the speed of light in a vacuum. It would actually have to interact with photons to observe them; and it is axiomatic that it does not.

As I said above, it doesn't have to be photons; it could be dark photons or gravitational waves, or just massive particles approaching the light cone as a limit. Maybe more to the point, though, it seems unlikely that dark matter forms structures organized enough that dark-matter-based life could exist, so it's not going to be doing any experiments anyway. We do the experiments. So far we haven't detected dark matter non-gravitationally, but if/when we do, the detection process will tie it to our established concepts of space and time and etc. Our gravitational detection of it already does, actually.
