Tapering makes no difference in Snell's law. If you have a refractive index profile n(z) [approximately independent of x and y], and you know the angle of the light at z1, then you can figure out the angle at another point z2 using Snell's law -- and the angle only depends on n(z1) and n(z2), not n(z) for any other z. It doesn't matter how smooth or sharp n(z) changes. (The profile and smoothness does affect the reflection and transmission amplitudes, but it does not affect the angle.)
The atmosphere is a good example -- the refractive index changes very smoothly from earth to outer space, as the air gets thinner and thinner, but there is still a sunset effect ... and moreover, you can calculate the sunset effect without knowing anything about the refractive index profile of the atmosphere. You only need to know the refractive index on the ground and the refractive index in space.
Therefore, if the refractive index changed at the edge of the solar system, we would see stars in weird, distorted positions, and the distortions would change over the course of the year, as the earth moved relative to the edge of the solar system. (Yes, I know the earth is kinda near the center of the solar sytem, but it still moves to some extent.) The relative positions of stars are measured to extraordinarily high accuracy, so if there were a refractive index change, it would be easily seen, unless the refractive index change were extremely small. I haven't gone through the calculation of exactly how small the refractive index has to be to be undetectable. Certainly a change of 1,000,000 can be ruled out!
-- ADDENDUM --
Here's a more specific way to think about whether it's astronomically observable. Just to be clear, let's do both 1:10^6 and 10^6:1 index contrast.
If the index is 1 inside the solar system and 10^6 outside... (Light travels slow in the interstellar medium.)
Then whatever direction we look, the rays exit the solar system more-or-less normal to the shell at the "edge of the solar system", by Snell's law. No matter where earth is, we see all the stars in almost exactly the positions they would be if we projected them onto the shell. (It would look like the ancient idea of a celestial sphere.) Then the question becomes:
Can we observe the parallax of an object at the edge of the solar system?
The answer is: Yes, very very easily. We can observe the parallax of stars far beyond the solar system, and we can observe that faraway galaxies have a parallax that's much smaller than that. So we can rule this possibility out.
If the index is 10^6 inside the solar system and 1 outside... (Light travels fast in the interstellar medium.)
Then almost any direction we look will have total internal reflection at the edge of the solar system -- we won't see any stars at all, maybe we'll even see a reflection of the sun -- except for the direction where we are looking almost exactly dead-on normal to the imaginary shell at the edge of the solar system. In that direction, there will be a little angular window where we will see all of the stars in every direction squeezed together. I think we can rule this out too!
Best Answer
The reason is that the light is the fastest possible way of transmission of information. Yes, before measuring one-way speed of light we have to synchronize spatially separated clocks and to synchronize clocks we need to know velocity of the one way speed of light.
But, we can measure two-way speed of light with a single clock.
To use the sun as a source of synchronization is the same so as to use an equidistant from clocks source of light. But, the moment they will see the sun can is very “subjective” and depends on one-way speed of light from the sun towards them, which can be different.
Let’s this laboratory moves relatively to this source of light and one-way speed of light is different leftward and rightward. When the beam comes to these clocks, you still have to assume what was the one-way velocity so as to set hands of these clocks. You still don’t know this one-way velocity and are free to assign it certain value. If you will assume, that one-way velocity was c, these clocks will be Einstein – synchronized, and measured one-way velocity will be exactly c, even if this synchronization is “wrong”.
It is well-known, that Einstein – synchronization is a special case of broader Reichenbach – synchronization, which is also self–consistent. Reichenbach synchronization allows anisotropic one-way velocities until two-way speed of light is c. For example, if in one direction it will be infinitely large in the other direction it will be very close to c/2.
Lorentz Ether theory (which is empirically equivalent to SR assumes, that the Earth moves in “preferred” frame, or Ether. Yes, according to this theory, as you say “light is NOT constant since we on earth are moving though space. & since we are moving though space we should observe speed of light as speed of light +- our speed of travel in the opposite direction.“ In Lorentz theory one way velocities of light are the same only in the Ether, but we cannot measure it because of well – known reasons. What we can measure, is two-way velocity, which is isotropic indeed, see Michelson Morley experiment. But, this experiment, as well as any other is not able to say something about one-way speed of light.
It is often said that one-way speed of light is conventional. So, time dilation in certain sense is also conventional, because it depends on synchronization procedure. If each observer synchronizes clocks by Einstein, they will see dilation of each other clock.
Let’s there are two relatively moving reference frames, S and S’. Let’s S’ moves very close to c. If clocks in these frames synchronized by Einstein, they will be “slower” each other.
If in the frame S the observer synchronizes clocks by Einstein and the observer in S’ synchronizes clocks by Reichenbach, assuming that he himself is moving in the frame S and one- way speed of light is anisotropic in his frame (since it is isotropic in S), this observer S’ will see, that clock S is ticking $\gamma$ times faster.
There are a couple of free sources that compare “isotropic one-way ” relativity and “anisotropic one” :
On the simplest examples of floating in water ships this article or this book simulates all kinematic effects of Special Relativity, Lorentz transformations, anisotripic one-way speed of light and isotropic two-way, time dilation, length contraction, relativistic velocities addition, Relativistic Doppler effect, reciprocity of Lorentz transformations, Twin paradox, Bell’s spaceship paradox etc.
Another was: Janssen, Michel (1995), A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble, but I was not able to find it right now.
Another well-known book: Max Jammer; Concepts of simultaneity: from Antiquity to Einstein and beyond - https://muse.jhu.edu/book/3280
Good to note, that the one way speed of light is anisotropic on rotating ring.
Does Sagnac effect imply anisotropy of speed of light in this inertial frame of reference?