Let me try and answer this question from a very theoretical point of view. But first, we need to set up some assumptions. The principle of relativity is a good assumption to start with.
Now, as the OP mentions, the aliens have set up devices just outside the solar system to jam us and prevent us from leaving. Fair enough. But can we sit inside our solar system and observe this? Probably not, because the aliens would not like us to find this out. So they might set up a system where these devices they have placed is basically a sphere around us (we are inside a Matrioshka brain?) , and they simulate the outer universe. Possible? Maldacena would like to agree with you. There is a theory that intuitively states that the description of a volume of space can be encoded on it's surface. Given the ability for instantaneous communication of the device outside our solar system, it can have a number of parallel processing components anywhere in the universe if required to fine tune a simulation for the tests that humans do. This means that no observations that we do without going outside the solar system (if possible) would not reveal the actual structure of the universe.
But what about local observations (observations withing the solar system)? No changes from our known results. Standard relativity applies, so all's well.
Fine, so the aliens can hide themselves till we venture outside. But all of this was deduced assuming that the aliens exist and can jam a bounded region. Let's talk about the second assumption fast. Looking at the structure of the universe right now, it's like this
where I would like to stress that this is a projection of the universe onto a 2-D plane, and that the outer universe might not be flat as shown, but may have a different structure.. The space inside the circle (region $R_1$ is our solar system), and the rest is obvious from this observation. Now, if there are any inconsistencies in a universe with a closed and bounded region having a different light speed than the rest of the universe, then it's bound to occur at the intersection as it is the point of discontinuity. To that end, how can two points, say $A$ and $B$ on the boundary communicate? Since the region bounded is convex (had it been concave, there would still be points that will lie in a locally convex region), the straight-line-communication would have to go through our universe, but then they would be limited by the speed of light of our region and also have a chance of giving themselves away. To avoid such things, they can use a external point $C$ and since $A\to C$ and $C\to B$ is instantaneous, $A\to B$ is also instantaneous.
This solves the problem as to what would be the speed of light on the boundary (if it makes sense, like using continuous refraction to keep it on the circle or something like that). How? Define a relation $\sim$ on the set of points $R=R_1\cup R_2\cup R_3$ (where $R$ is the whole universe) such that for $x,y\in R$, $x\sim y \iff x$ and $y$ can communicate instantaneously. Observe that the relation is trivially reflexive and symmetric ($x\sim x$ and $x\sim y\Rightarrow y\sim x$) and by the argument in the above paragraph, $x\sim y,\ y\sim z\Rightarrow x\sim z$ and thus the relation is transitive. This shows that this an equivalence relation and hence $R$ is partitioned into 2 subsets $X,Y$, the former where all points can instantaneously communicate, the latter cannot. Since $A,B\in R_3$ can communicate instantaneously and since $A,B$ was arbitrary, $R_3\subset X$ and thus $X=R\setminus R_1,\ Y=R_1$.Thus light speed is infinite (in a mathematical sense) on the boundary.
So far, no inconsistencies.
Now we ask the following question. How will two particles, one in $R_1$, the other in $R_3$ communicate?
We first look what happens if they are electrically charged. This won't be a problem because of electromagnetic shielding. The aliens can hide any traces of them, electrically. But what happens if one looks at gravitational effects? Can these devices hide geometrical distortions in spacetime? Hard to say. If there is a particle in $R_3$ that bends spacetime in a way that we can detect using the particle in $R_1$, then the detectors might take in account the effects and simulate a visual program for us to see to validate our observations but giving no information about the outside universe. Possible I suppose, but very very difficult. Might be closer to impossible, but here I run out of maths in this regard.
So let's take it that they can keep us in the dark even though we try to measure outside stuff via things that are globally affected.
Then, is there any known theory that stops them from doing so> The Copernican Principle? Not so much a problem, because it's a local issue and happens in a closed and bounded region (I can say compact, but let's go with this). Because suppose you throw out $R_1$(mathematically). Does $R_3$ have a problem with that? No, because all points in $R_3$ can communicate instantaneously and hence communication-geodesics (the line of fastest communication) can circle round the hollow region without any problem. There might be some physical issues, but none fatal enough. This does break homogeneity though. But one can characterize the hole as a singular region, akin to a black hole and shut physically it out by not communicating with it. There will be residual information leakage and other effects, but nothing inconsistent that would cause a logical collapse.
One might observe that no information about $X$ ever reaches $Y$ and hence $R_3$ is a horizon, and hence there should be a radiation quite similar to Hawking or Unruh radiation (random particle emission), and we can observe this to know that something is wrong, and there exists a horizon. This line of argument does not work, because we can't observe this, as we would be living inside, roughly speaking, an inverted black hole. The leak would be outside, into $R_2$.
Think as I may, I cannot see any reason why the aliens cannot jam a bound region and not give themselves away. This leads us back to the assumption that the aliens exist.
The existence of the aliens is an extension to the question of whether a self-consistent universe with an infinite light speed exist. Possibly... yes. Well, at least, the following provide evidence for such a universe to be consistent.
Assuming the principle of relativity, and that the universe is isotropic and homogeneous, Sir Palash B Pal tells us that two kinds of relativistic systems are consistent. One with a constant $K=0$ and the other with a constant $0<K<\infty$. He argues that the first case arises from the assumption that time is absolute, which is basically a universe where Galilean invariance works ($X$)and the other is basically our universe $Y$. Until we go poking around $R_3$ (when we will obviously detect the presence of a sphere), $R_1\cap R_2=\phi$, and $R_1$ is closed and bounded and hence is consistence.
How shall the dynamics of such an universe work? Pretty much just classical mechanics, non-relativistic QM, and the like. Electromagnetism is inherently relativistic, but non-relativistic electromagnetism, the so called Galilean electromagnetism, exists and is consistent with instantaneous light speed. Interestingly, spin-orbit coupling goes out the window in such a universe. A rather boring set of atomic orbitals. What about gravity? Gravity in $Y$ has a geometric structure, which can be explained by General relativity. But in $X$? Fortunately, there is a consistent geometric formulation of geometric Newtonian gravity, called Newton-Cartan theory. Again, the geometry in $X$ and $Y$ has to agree (and be differentiable) at $R_3$, which the devices probably will have to do if they want to avoid detection. Is it mathematically possible? Yes. The reasons are involved, but it works.
Can such a universe have a beginning? Hard to answer, possibly yes? But the birth of stars and everything would have to be modified. The existence of mass-energy equivalence is purely relativistic, but strictly speaking, such a thing cannot be said to be inconsistent with classical physics. One can insert such a term and get away with it because all it would do would be to shift the energy scales from 0 to some other value. But would it have any physical meaning? No. An example to why this is so is simple. Take a particle of some mass. Let it radiate two photons, one to the left and the other to the right, both of equal wavelength. Let's see what $X$ and $Y$ can deduce. First in $Y$. The center of mass frame of reference says that the particle stays at rest because both the photons carry equal momentum. But in a moving frame, one of the photons would have higher energy, and hence higher momentum and thus the body would have lost some momentum. To reconcile both the observations, they deduce that the body has lost some mass. But in $X$, wavelength does not make sense. All radiation are instantaneously propagated and hence the word loses its meaning. Both the photons have the same energy (it it makes sense) and hence same momentum in whatever frame they are observed and hence makes no deduction about the particle's mass or energy. In other words, stars possibly cannot exist.
First line of thought that can produce an inconsistency found! If no stars are formed, then no higher elements as there would be no nucleosynthesis, and hence no life. I do not believe life can be formed with just the lower few elements, and pure consciousness is not something that can be defined with current theories that we have. Is this enough to kill off the aliens? Highly probable, but there can be one way (more than one way) the aliens can still crop up in such a bleak desolate universe.
A stabilized rotating Kerr black hole! Or something similar.
Let's suppose the aliens are a Type-III or Type-IV civilization (likely the latter) residing in a universe similar to ours. Maybe something similar to the extraterrestrials in Arthur C. Clark's 2001: A Space Odyssey. They have discovered a theory of how to jam regions of spacetime in a Galilean universe to induce a Einsteinian universe in such a place. They also have the technological know-how, only no way to test. They also have the potential to open a portal and pass on their information to other universes in the multiverse. One such thing reaches a universe that looks like $X$, a Galilean universe. They see no life, just bleak primitive elements and compounds spread across a desolate, barren and dark landscape. The aliens reconstruct themselves in this universe, and then start constructing their device, on a large scale, which, according to them, would encompass several thousands of galaxies in their universe. Once done, some of them stay inside the structure, and the rest leave the universe for other work.They generally don't disturb the device, except to look at what the situation is on different places inside the region. Due to relativity now doing it's work, stars form, heavier elements arise and a miniature Einsteinian universe starts. Ages pass, and life births on several planets, one of them being Earth in the Solar System (their names are different to the aliens, but we call them so for easy identification). After several million years, intelligent life starts on Earth, none of the others being fruitful. The aliens are happy with their simulation, and they construct a similar device just around the solar system. After completion, they deconstruct the outer device.
This obviously puts us in the situation the OP has right now, albeit not the process the OP was probably thinking.
I don't have anything to add right now, but if I do find anything else, I would add them up.