> edge of my understanding and a bit beyond
...
> Are bell inequality violations Lorentz-invariant?

Bell tests are tests of local realism, so no, not exactly.

Local Lorentz invariance guarantees local causality, which means that objects can non-negligibly influence each other only when they are in close proximity in spacetime. If we slice up spacetime into space and time, then objects together in a small test apparatus may give evidence of a certain interaction, whereas an object in the apparatus is not going to give similar evidence of interaction with an object in a laboratory across town. Likewise, if we put object one into the apparatus, then remove it, wait some time (minutes, months) and put object two into the apparatus, we won't expect evidence of interaction between the two to be registered by the apparatus.

Local realism is that concept of local causality combined with realism in the sense that objects can have persisting properties (over time) and that objects too far away to measure nevertheless "do" have persisting properties. The first is a "timelike" separation, the second can be "spacelike separation" at spatial distances so large that the travel time of light becomes impossible to ignore.

Achieving and proving spacelike separation is a frequent goal of Bell test experimenters. Usually this means that two objects interact and acquire a special property which persists as the objects are moved apart (in space and/or time) whereupon this property of each is examined by an apparatus that is forbidden to communicate the result of one examination to the apparatus doing the other examination. The apparatus or apparatuses must have no memory of whichever result happened first. The property is special in that the objects, when examined, show "opposite" values.

The intriguing issue is that results of the examinations of the entangled objects looks a lot like there is some memory or communication involved. If one apparatus, maybe the apparatus is somehow changed by the first examination in a way that determines the second examination; if two apparatuses, perhaps the first apparatus communicates its result to the second; if the two apparatuses are too far apart to communicate the result of the first examination before the second examination is made ("too far"
limited by the speed of light), then perhaps it's the objects that have the memory, perhaps the entire universe somehow remembers, or perhaps the speed of light is not universal (which would be a clear Lorentz invariance violation). (There are other options available).

Clearly pinning down just a violation of local realism is hard enough -- over the past four decades there have been hundreds of different types of experiments attempting to prove local realism violation. One weakness is that entanglement decays with exposure to the environment, and isolating each entangled object during transport (or storage, if doing a separated-by-a-long-time experiment) is very difficult. The workarounds for this often muddy understanding of the results. But even as experiments are strengthened, they might not prove a violation of the principle of locality rather than that the combination of that principle and the principle of realism don't fit together in all circumstances. So while a Bell inequality violation is suggestive of a Lorentz-invariance violation (LIV), taking that to proof of LIV is tricky. (For example, the states, the Bell operator and the Bell observable do not need to be individually Lorentz-invariant when predicting a measurement outcome probability, and proving whether they individually are or are not is much harder than proving that the combination of these states and operators is (or isn't) Lorentz-invariant).

Semi-jokingly, perhaps the best way to prove a LIV with a Bell test is to have the scientist and half the entangled objects step into a faster-than-light (FTL) transporter which whisks them to a laboratory on Mars before the scientist and her collaborators can blink their eyes, then Earthbound-collaborators and on-Mars experimenter promptly perform the tests knowing they can prove the tests were at a spacelike-separation through e.g. during-experiment astrophysical observations (but that confidence arises because of the performs-measurement-while-on-Mars scientist's explicit FTL travel, and FTL travel is LIV in itself).