Is it possible to prove or disprove the Einstein Synchronization Convention with astronomy?
Your question is really about Einstein's second postulate, the constancy of the speed of light, not about the synchronization convention, which is only a convention as the name implies.
Light moving at a constant speed has consequences that are independent of what that speed actually is. It means that light waves won't overtake each other, no matter how far they travel. We can test this in various ways. For example, binary stars and moons in the solar system accelerate over fairly short time scales; if there was even a slight dependence of the speed of light on the speed of the source then we would see distortion in their motion that we don't see.
Once you've convinced yourself that the speed of light is constant (in that sense) by these sorts of observations—which had already happened before Einstein's paper—you can choose to use this property of light to set clocks. If the speed is really anisotropic then clocks that you set this way won't really be synchronized, but that doesn't stop you from setting them this way. You can now ask another question: if you set clock B from A this way, then set C from B this way, is the result the same as if you'd set C from A? You can test this by setting two different clocks at C and comparing them locally. You can repeat this experiment with every possible arrangement of three points at relative rest in three dimensions.
If the speed of light passes that test, then it no longer matters whether it's "really" isotropic or not since it behaves as though it's isotropic. We can assume our clocks to be synchronized, and we can even fix the speed of light in meters per second by definition and use it to define length, as we in fact do. This doesn't prevent us from detecting a violation of our assumptions, because the only physically meaningful assumption that we actually made is that the experiments of the previous paragraphs won't start returning different results in the future, and we didn't assume the existence of "truly" synchronized clocks for those experiments.
We can also take the speed of light to be anisotropic, and the clocks to not be synchronized, but this amounts to doing the same physics in different coordinates, and the result of any calculation in these coordinates will be the same as the transformed result of the calculation in standard coordinates. Taking the age of stars as an example, if the $t$ of the anisotropic coordinates doesn't match cosmological time, then stars at the same distance in different directions have different ages, and this exactly counters the light travel time delay so we see them at the same age. If the $x$ coordinate doesn't match comoving position, then Earth is moving away from the faster light and toward the slower light at just the right speed so that they take the same time to arrive. If both coordinates don't match, it's a combination of both effects. This is similar to the way that length contraction, the relativity of simultaneity, and so forth always conspire to make things consistent in different inertial frames.
if the Einstein Synchronization Convention is untrue, and there is a directional difference in the speed of light, we should be able to notice this through astronomy; if the light takes less time to travel to us from some directions than others, we should see older stars and galaxies in the directions where light travels more quickly than in the directions where it travels more slowly
This is a good question. Neglecting the CMB dipole anisotropy, we see a very nearly isotropic large scale structure to the universe. So your question is, how could a non-isotropic synchronization convention possibly explain the observed isotropy?
As you say, light from the “fast” direction would have a shorter delay than light coming from the “slow” direction. So the fast light would give more recent data and the slow light would give more ancient data. Since both directions show galaxies of roughly the same age, that means that there is an anisotropic cosmological gravitational time dilation. Galaxies in the fast light direction age more slowly and galaxies in the other direction age faster.
Yes, such a convention would be very cumbersome and inconvenient, which is why it is not used. But it would be self consistent and also consistent with the cosmological data.