Well, first, it bears pointing out, there's a fundamental problem with calling any dimensioned quantity constant or variable: the infinite regress implied. What about its units? And with respect to which units do you assess their constancy or variableness?
How do you do you even compare the units at one place and time with those at another place and time, except by moving them over from one place to the other and just asserting by decree that it's the "same" unit? In fact, how can you even define derivatives for dimensioned quantities, if you have no way of directly comparing units at different places and times?
So, the very terms "constant" and "variable" are meaningless; and the answer is that light speed is neither constant nor variable, but is a dimensioned quantity.
So, to by-pass this issue it is better to both deconstruct what we're actually trying to say with the central hypothesis of Relativity (both Special and General) and rephrase the question; doing so in a way that will allow us to actually resolve this matter at a much deeper level, and better express what key postulate of Relativity is actually saying.
Relativity: Contrasted With The Newtonian and Carrollian Worlds
The central postulate of Relativity, both Special and General, is that space and time - together - form a chrono-geometry that locally looks like a Minkowski space. The key property of Minkowski space is that at each point in space and time, in each direction, there is a finite non-zero speed that is absolute.
Contrast and compare this with other conceivable possibilities. In the world of Newtonian physics, the assumption is that instead of a finite speed, it is the infinite speed that is absolute. This is the speed of being at different places at the same time. So another way of expressing the same idea, through the back door, is that the quality of being simultaneous is absolute. That's referred to as Simultaneity.
Another, lesser discussed, possibility is a world in which the absolute speed is 0. That's referred to as the Carrollian universe, because in it, things can have momentum, but don't move anywhere.
Another possibility is where all 4 dimensions are spatial and there is no time-like dimension nor any notion of "speed" at all; just a timeless 4D space. There, instead of "speeds", you'd be talking about "angles" with respect to the 4th spatial dimension.
The Temporal Logics Of Newtonian and Minkowski Worlds
In the world espoused by Newton, there is such a thing as a "there and now". For instance, with respect to any place and time on Earth, there is a unique time at any given location on the moon that is "now". All times after it are in our future, and all times before it are in our past. Likewise, with respect to that "now" on the moon, all points after our now are in its future, and all points before our now are in its past. What is past and future of a given time is, thus, independent of your location.
In a world that is locally Minkowski, there is no such thing as a "there and now". In particular, infinite speed is not even absolute, but relative. With respect to a moving frame of reference an infinite speed transforms to a finite faster-than-light speed. Conversely, for every faster than light speed, there is a moving frame of reference in which it is infinite.
Correspondingly, with respect to any point on Earth at any time, since the moon is about 1½ light seconds away (using very round numbers), there is a gap of about 3 seconds' worth of the moon's time line that can be considered to be "simultaneous" in some frame of reference. As such, no part of that gap can be considered to be either before or after here and now on Earth. It is neither in our past nor our future. The term sometimes used is "absolute elsewhere".
Another example: Betelgeuse is (using round numbers again) about 1000 light years from us. If there is more than 1000 years remaining before the view of the explosion of the star reaches us, then the explosion is in our future. If there is under 1000 years before we see its explosion, then it is neither in our past nor our future.
The notion that "it has already happened, but we haven't seen it yet" is wrong. That's Newtonian thinking, and even some astrophysicists (like Tyson, hello Tyson?!) have been sighted talking Newtonian. In the world just described here, the explosion still hasn't happened yet - in our past. There is no "now" or "just now over there, but we haven't seen it yet", because there is no such thing as "there and now" to begin with.
That talk is qualified a bit, further below, with the mention the "co-moving" frame, and that's the only out someone like Tyson could claim. As an aside, I've also seen Sabine Hossenfelder employ the co-moving frame, in one of her videos, as a possible way to define an "absolute now". But this is all pertaining to General Relativity, not Special Relativity.
From 3D Equal-Time Layers To Light Cones
If you were to draw a diagram, suppressing one spatial dimension, showing the everyone's time line in the vertical direction, the locus of all light-speed trajectories emanating from a given point in space and time, would form a conical shape. That's called its "future light cone". Similarly, the locus of all light-speed trajectories arriving at that point in space and time would form its "past light cone".
The past of the point at the given time lies on and within the past light cone. In particular, your visual field - including the sky - is on your past light cone and, for all intents and purposes, is your past light cone.
The future of the point at that time lies on and within the future light cone. The rest of the 4D continuum is the absolute elsewhere.
What's your "past" and "future" depends on both when you are and where.
So, it is not that "light speed is constant" in Relativity, but that it is giving you an infrastructure of a world that's different from that espoused by Newton. Instead of a stack of 3D "equal time" spaces layered up one on top of the other, as is the case in Newtonian physics, you have a 4-dimensional continuum enmeshed with a field of light cones.
Constancy Of Light Speed: Special Relativity
Now, we can address the matter at hand in a much more direct way that gets to the root of the matter. Instead of asking "is light speed "constant", in its place, we can ask: "is the field of light cones constant"?
At bare minimum, this should consist of at least the following:
(1) The field of light cones is position-independent; i.e. spatially homogeneous.
(2) The field of light cones is time-independent; i.e. temporally homogeneous.
In that way, we can assure that it is the same at all places and times.
Second: we also want it to be the same in every direction, so that:
(3) The field of light cones is isotropic; i.e. invariant with respect to "rotations"; i.e. reorientation of spatial axes.
Finally: the key hypothesis of Special Relativity is that light speed be observer-independent. So, we also want:
(4) The field of light cones is invariant with respect to "boosts"; i.e. changes to moving frames of references - those changes, in particular, being identified as "Lorentz transformations".
Generically, a transform to a moving frame of reference is referred to as a "boost". The contrast being drawn here is between the Lorentz transform and the one applicable to the world of Newton - the Galilei transform.
Under Galilei transforms, the layering of the world into 3D "equal time" slices remains fixed, but not the field of light cones (nor any field traced out by finite speed trajectories).
Under Lorentz transforms, it is the light cones that remain fixed, while the layering into "equal time" 3D slices does not.
So, here's the conclusion: a chrono-geometry that possesses a field of light cones, along with a set of transforms for (1) spatial translations, (2) time translations, (3) rotations and (4) boosts that leave it invariant is one and the same as a Minkowski Geometry. It is one way to characterize a Minkowski Geometry.
On a slight technicality, it's not completely true, because you also have conformal Minkowski spaces, but I'm just going to ignore the case, for the sake of expediency.
Non-Constancy: The Generalization To General Relativity
If "constancy" is defined this way, then the transition from Special Relativity to General Relativity - a transition away from Minkowski Geometries to more general "Lorentzian space-times" consists of rescinding the constancy condition.
Instead, the assumption reverts to the weaker condition that the invariant speed everywhere be finite and non-zero, while allowing for fields of light cones that need not be spatially or temporally homogeneous, isotropic or boost-invariant.
Contrast Minkowski geometry to the geometries presented in General Relativity. In particular, for any of the Big Bang geometries (i.e. the instances of the FRWL metrics), while it is true that we have a field of light cones that possess (1) spatial homogeneity and (3) isotropy, they are not invariant under (2) time translation, nor under (4) boosts!
The lack of boost-invariance means that a particular frame is selected out. In cosmology it is referred to as the "co-moving" frame. With respect to it, the Earth has a definite motion ... I think it's in the direction of Sagittarius.
In the co-moving frame, the Cosmic Microwave Background would appear color-uniform. In contrast, in the Earth's frame, there is slight red-shifting on one end, and slight blue-shifting on the other.
Another property of the co-moving frame - far more important - relates to the question of the size of the Universe. When we say "the universe is spatially finite/infinite" - which 3D layering are we referring to? And the answer is: the 3D layering that is associated with the co-moving frame.
Technically, the layering is the locus of all spatial paths that are "orthogonal" to the co-moving trajectories. It is only in the co-moving frame that this layer wraps around to form a connected 3D layer (that is, if the Universe is spatially finite). In all other frames, the corresponding 3D layer (for the spatially finite case) would wrap around and come back to our location at a different time, thereby coiling to make up a infinite 3D helix.
The lack of time invariance means that there is time dependency on the structure of light cones. One way of saying this is that the speed of light is simply variable, and that this is the variability that the FRWL metric is actually describing!
But no matter how you describe or characterize it, the fact remains, the field of light cones for the FRWL metrics do not possess time-translation symmetry.
We can see this more clearly as follows. The FRWL metrics allow for 3 main cases: (1) where the spatial 3D layers are uniformly positively curved and wrap up into hyperspheres - the above-mentioned "spatially finite" case, (2) where the spatial 3D layers are uniformly negative curved and form an infinite hyperbolic geometry and (3) where they are flat and form a 3D Euclidean geometry.
Hence, the origin of the joke "define the Universe and give 3 examples".
To a very high degree of accuracy, the observed universe falls into the the flat case (3). For such cases, the FRWL metric has a form given by the following line element:
$$A(t) \left(dx^2 + dy^2 + dz^2\right) - c^2 dt^2,$$
where the "growth" factor $A(t) > 0$ varies with time, in such a way that $A(t) → 0$ as $t → 0$, where $t = 0$ is set as the time of the "Big Bang".
For Minkowski geometry, $A(t) = 1$, and $c$ is the finite non-zero absolute speed that is the topic of this discussion: the "(in vacuo) speed of light".
However, this way of writing the metric - as a line element for distances - obscures the view of the matter. If you rewrite it as a line element for proper time
$$dt^2 - \frac{A(t)}{c^2} \left(dx^2 + dy^2 + dz^2\right).$$
Now, you can more clearly see that this metric:
$$dt^2 - \frac{1}{c(t)^2} \left(dx^2 + dy^2 + dz^2\right).$$
that is just that for Minkowski space, but with a variable speed of light:
$$c(t) = \frac{c}{\sqrt{A(t)}}.$$
In turn, this also helps make more clear just what the Big Bang "singularity" actually is. The $t = 0$ hypersurface of this geometry is actually an envelope of light cones. Another way of saying the same is that it is a "null surface"; and that all paths in the "t = 0" surface are "light-like". In other words, light speed is infinite on it and it's a 3D slice of Newtonian geometry! (That is: of a Newton-Cartan space-time).
Technically, the "t = 0" surface actually violates the "local Minkowski" axiom for General Relativity, if it's included in the geometry. Therefore, most cosmologists exclude the "t = 0" surface and treat only the "t > 0" sub-space as the Big Bang space. There are others, like Hawking & Hartle people, or the people who hang with Mansouri, who consider and treat "signature changing" geometries. Hawking's treatment still excluded the "t = 0" surface, and he had a quantum jump from the "t < 0" subspace, which is purely spatial 4D locally Euclidean and is not a space-time at all (a "timeless space") to the "t > 0" subspace, which is the locally Minkowski space. Others, like Mansouri, also include the "t = 0" null 3D surface and try to address the singularity issue head-on.
So, for this kind of geometry, it is only the "t > 0" subspace that conforms to the "locally Minkowski" axiom of General Relativity, and you don't even have that - much less the constancy of light speed axiom, if considering the entire geometry for all $t$.
