"The Bell Inequalities are focused on causality of quantum events that are space-like separated whereas the focus of this thought experiment is simultaneity between frames that are space-like separated."
That's what I was trying to say. Simultaneity is only a problem if it violates causality, otherwise you can still argue that things are relative. The causality problems of Bell's paradox arise from the simultaneity issue, so in that way it's essentially the same. According to Einstein's original statements on relativity, the example given IS a violation. However, the symmetry of the situation allows one to have an out, leaving relativity primarily intact.
"Why not also consider the formulation (i.e., postulates) of special relativity?"
That's what I was doing, but I'm just saying that it's not a major blow to relativity. It turns out that QM remains completely untouched with that solution, but relativity must be SLIGHTLY modified (just from a conceptual point of view) from Einstein's original intention.
If you think about the situation you presented, there is absolutely no useful to propagate mass OR information with it. Thus, one can argue the speed of light is exceeded, but not in any meaningful way.
Imagine if you could send messages at speeds greater than that of light. Let's say I sent a message informing you of the death of a friend of yours. Let's also say that you respond by saying a prayer for his or her soul. According to relativity, there is some frame in which that prayer would occur before you received the message. This arises from only simple space-time arguments, not some obscure facet of the theory. This is why physicists tend to be so insistent about the speed limit thing. We are not gods, however, so there is no guarantee that we have captured every facet of the situation. I'd really like to go faster than light too, but I have trouble seeing how it could be done.