I don't currently have access to the full text of anything that Penrose wrote about the "functional degrees of freedom"; but back when I did, I couldn't see his point, and searching the literature now, I see no sign that anyone else understands what his argument is.
Using a notation he attributes to John Wheeler, he calculates the number of "functional degrees of freedom" e.g. in the magnetic field to be "$\infty^{3\infty^{3}}$". The idea apparently is that there are "$\infty^{3}$" possible values of a 3-vector (because it has 3 real components, each of which has "$\infty$" possible values), and there are $\infty^{3}$ points in 3-dimensional space at which a magnetic 3-vector must be defined, so the total number of possible configurations of the magnetic field is $\infty^{3}$ to the power of $\infty^{3}$, or, $\infty^{3\infty^{3}}$.
This is a method of counting degrees of freedom that I have never seen anywhere else, and moreover, it would seem to radically overcount the number of possible magnetic field configurations, if one takes the equations of motion into account - since most of those configurations won't even be continuous, let alone differentiable, and yet one needs these properties for the equations of motion to be well-defined.
In any case, what does Penrose want to do with this concept? Here I rely on
"On functional freedom and Penrose's critiques of string theory"
a recent paper from Oxford which is the only serious attempt I can find, to make sense of Penrose's arguments, and engage with them.
As catalogued in the paper, Penrose uses FF counting to
question holographic dualities like AdS/CFT (since they assert the equivalence of two theories existing in different dimensions, yet they should have a different FF count)
question the possibility that a higher-dimensional theory could ever give rise to the four-dimensional physics we observe (since the extra FFs should somehow dominate the dynamics)
There is also
mentioned in their section 4.3, which does not rely on FF counting per se, but rather applies a singularity theorem of the kind he proved with Hawking (and for which he got the Nobel in 2020) to the compact extra dimensions of string theory.
The Oxford authors argue that the first FF counting argument is invalid in section 2.3, and the second FF counting argument is also invalid, at the end of page 18.
As for the instability of string compactifications, this is actually a long-standing topic in string theory, in areas like "moduli stabilization" and the possible instability of De Sitter space and Anti de Sitter space. Even though these involve quantum superpositions of geometries, rather than the purely classical geometries of the original singularity theorems, it is conceivable to me that the classical Penrose-Hawking instabilities might have a quantum counterpart (similar to the way that classical chaos has a counterpart in quantum chaos). My comment here is simply that one should not expect string theorists to have a professional blind spot that would prevent them from noticing it, since they already discuss the stability issue on many other fronts.
A further general comment: stability or instability of dynamical systems is a vastly studied topic in mathematics. The Millennium Problem about the Navier-Stokes equations, for example, is a kind of stability problem. And yet I am not aware of this counting of "functional degrees of freedom" being used anywhere, except in these vague arguments due to Penrose.
So there's every chance that there is simply no substance to these particular objections from Penrose - at least those based on FF counting.
The general topic of instability is known in string theory. If we focus on this part of your question - could "instability of additional dimensions" have something to do with dark matter... The problem is that, if you had a string theory model that explained the known forms of matter, and then discovered that it had an instability, that would probably just wreck the whole thing, rather than add an extra new phenomenon while leaving quarks and leptons and atoms undisturbed.
Nonetheless, I suppose it is logically possible that a quantum Penrose-Hawking instability (if there is such a thing) could produce e.g. novel topological defects in the extra dimensions, with the extra dimensions being stable once the defects are produced, and that these defects could then be the dark matter. Evidently the key thing to investigate is whether the Penrose-Hawking instability does have a quantum counterpart. That is a question of interest, even without the extra speculation that this is where the dark matter comes from.
A few final comments. The size and shape of the extra dimensions and other geometric parameters, are known as "moduli", and there are models in which the dark matter comes from excitations of the moduli. There are also models in which dark energy is due to a single modulus slowly increasing (e.g. the radius of one of the extra dimensions), which is a kind of instability explanation, though not what you were talking about.
