Could speed of light be variable and time be absolute? I get my "demonstration" of time dilation from the textbook thought experiment.
A laser is mounted on a cart with a reflective ceiling. At $t=0$ the cart starts moving and the laser is fired. When the laser is reflected back at the starting point the (thought) experiment stops.
Now, two different observers, one sharing the frame of the cart and another standing on the ground perpendicular to the cart will observe two different things. For the first one, the laser bounces back and then down in a straight line. For the second one, the light travels in a triangular pattern which is longer than the path observed by the first guy.
Given that the speed of light is constant, the time has to dilate/contract.
Why is the speed of light held constant here? Could we work out a physics where time is absolute but the maximum speed of light variable?
 A: Keep in mind that several other Einsteinian effects are hard to explain in an absolute-time scenario and are tested. For these purposes I would concentrate on:


*

*The existence of a speed limit for massive particles  
Looping accelerator systems based RF cavities only work because once the particles have enough energy their speed is effectively constant (and that speed is within a hair's breadth of $c$). But this constant speed doesn't mean constant kinetic energy or momentum: the accelerator continue to add energy and momentum with measurable consequences in the bends and the experimental halls.

*Conversion of other energies to mass and vice versa 
We can measure the kinetic energy and mass of reaction products in particle physics experiments and when we produce heavier products than we started with the extra mass is related to lost kinetic energy in keeping with $E = mc^2$. Likewise when particles decay to lighter products the products have extra kinetic energy in keeping with the mass difference.

*The twin paradox  
It is not just a fanciful notion you find in books, but something that we do with unstable particles in looping accelerators (example, muon-g2). 
To fit all of those that into a absolute time framework will require more (and to my mind very arbitrary) assumptions. By contrast the Lorentz symmetry of special relativity explains them all in one go and is motivated by the structure of Maxwell's equations (that is has an experimental basis and is not at all arbitrary).
A: According to Lorentz Ether Theory simultaneity is absolute and one – way speed of light is frame dependent. One-way speed of light is isotropic only in the preferred frame, or Ether; hence in all moving in the Ether laboratories one-way speed of light is anisotropic; but introduction of length contraction and time dilation for all phenomena in a “preferred” frame of reference which plays the role of Lorentz immobile Ether leads to the complete Lorentz transformation. 
Because the same mathematical formalism occurs in both, it is not possible to distinguish between Lorentz Theory and Special Relativity by experiment.
For example, this paper simulates all kinematic effects of the special theory of relativity on the simplest example of floating on water ships.
When using the term 'the speed of light’' it is sometimes necessary to make the distinction between its one-way speed and its two-way speed. The "one-way" speed of light, from a source to a detector, cannot be measured independently of a convention as to how to synchronize the clocks at the source and the detector. What can however be experimentally measured is the round-trip speed (or "two-way" speed of light) from the source to the detector and back again.
A. Einstein chose a synchronization convention (Einstein synchronization) that made the one-way speed equal to the two-way speed. 
As soon as observers in relatively moving frames employ the same (Einstein's) synchronization scheme of clocks they would make different conclusions whether  spatially separated events occur at the same time (relativity of simultaneity).
It should be noted, that Earth rotates; hence one-way speed of light relatively to the Earth surface and any laboratory that is on the Earth surface is different in different directions (or anisotropic, see Sagnac Effect) while two-way speed of light is isotropic (see Michelson Morley experiment).
A: There are a number of alternative theories to SR that posit absolute simultaneity (and hence reject SR's relative simultaneity).  There are often referred to as Lorentzian relativity or neo-lorentzian relativity, and were originally developed by Lorentz and others (such as Poincare).  They preceded SR by approximately 10 years.
These theories presuppose that there is only one frame (the "preferred frame" often called the ether or ether frame) in which the speed of light is truly isotropic.
In these theories, time is absolute and does not vary with speed.  Here it is theorized that there is no actual "time dilation."  There is only "clock retardation."  Time, as an abstract concept, does not change.  However the mechanical instruments which we use to measure time (clocks) are altered by motion.
According to such theories, the speed of light is variable, and differs from frame to frame (while time does not).
However, the speed of light is always MEASURED to be C in other frames if corrections are not made for the distortion of the measuring instruments caused by increased velocity.
The idea is that all (uncorrected) frames measure C to be the same because the rods and clocks used in those frames have been distorted by velocity.  So, even though the MEASURED speed of light is C, the "actual" speed is not (except in the preferred frame).
These theories are often said make the same predictions as SR (although I would dispute that).  Therefore, every experiment which is said to confirm SR can also be said to confirm LR.
Sexl and Mansuri did a several part, extended analysis which compared and contrasted SR with such alternative theories.  Here is an excerpt from a wiki article which summarizes their work on the topic:
"So Mansouri and Sexl spoke about the "remarkable result that a theory maintaining absolute simultaneity is equivalent to special relativity." They also noticed the similarity between this test theory and Lorentz ether theory of Hendrik Lorentz, Joseph Larmor and Henri Poincaré."
https://en.wikipedia.org/wiki/Test_theories_of_special_relativity
The essence of your question was:  "Could we work out a physics were time is absolute but maximum speed of light variable?"
The answer is "yes, we can (and have)."
Edit:  I see now that I am basically just re-stating, in a different form, the answer given by Albert, above.
Edit:  The second part of your question was "Why is the speed of light held constant here?."  Because it is a (by definition unproven) postulate of SR.  Einstein himself readily acknowledged that it was a "freely chosen" premise, not one that was dictated by experimental evidence.
One the other hand, Lorenztian theories posit the (also unproven) premise that the speed of light is not "truly" constant in every inertial frame, even though we measure it to be the same.
A: There is a physics where time is constant and the speed of light is not: Newtonian with an ether.
Galilean transformations in Newtonian physics have time as a fixed parameter that ticks away uniformly for all points and states of motion.
For the speed of light to be frame dependent, one needs the ether in which it propagates at $c$. The ether then defines a preferred reference frame against which all motion is absolute.
So in the train experiment, the station (which is solidal with the ether ;-) sees light take a long path at $c$. Meanwhile, the observer on the train sees light propagating vertically more slowly:
$$ c'  = \frac c {\sqrt{1+(\frac v c)^2}}$$
Now if you want to find a theory that is fully Lorentz invariant such that it correctly matches the train-light-clock experiment and the other standard S.R. scenarios, then: No. Relativity is the Lorentz invariant solution.
A: No.  You're neglecting that time dilation is only one effect explained by relativity.  You'd also need to take into account length contraction or, more generally, Lorentz invariance.  Your "absolute time" is not going to capture the fact that there are properties of spacetime (not just time) at play here.
In light of the comments (pun intended!), a further point:  The idea that the speed of light is constant is a consequence of the Maxwell equations.  Relativity follows that logically and historically, it doesn't precede it.  If you open the question to spacetime, then you still have the issue that the Maxwell equations are and were well-known.  Special relativity is about exploring the consequences of that experimentally verified fact.  It's not clear what you mean by "absolute" here, but any interpretation that comes to my mind implies a preferred frame for Maxwell, and we know that does not exist due to experiment.
A: 
Why is the speed of light held constant here?

You are missing one important point of that thought experiment: it is meaningful after considering the Michelson-Morley experiment.
If our interpretation of the Michelson-Morley experiment is that the speed of light is constant with respect to the observer, then we can perform the thought experiment you described, using the assumption that the speed of light is constant (we justify this assumption because of the results of the Michelson-Morley experiment).
That thought experiment leads then to the formulae of time dilation and space contraction (and to the special relativity theory).
A: When you consider different frames, something weird happens.
Frames result in you measuring things in weird ways that cause problems for a Newtonian explanation of the world, with a single cartesian space with galilean transformation between frames, and a single universal time.
The question is whether SR is the only way to resolve that. 
Any alternative would have to give the same results as SR, so I will revise the question:
Is there any way to think about it that allows a single cartesian space with galilean transformation between frames and a single universal time, that is equivalent to SR?
To get that, we would have to somehow pile all the weirdness into the motion of electromagnetic force through space.
Here's a link to some graphics which display part of the problem, and propose a constraint that any potential solution must meet.
https://www.glowscript.org/#/user/jethomas5/folder/LW3/program/galileanrelativity
At a minimum, to save cartesian space and absolute time, then when light or EMF travels from a source charge to a target charge, the speed must vary just the right amount to get constant lightspeed $c$ relative to the target.
So two charges at the same location with different velocities would observe the same light at different times.
This is so weird. Much weirder than SR, and probably it can't be equivalent to SR. So probably there's no way to do it with cartesian space and a single time.
