Let's see if I can do this argument justice...
Just exactly why is this the case? "But since the measurement events are space-like separated,
there will also be an inertial frame Frame(F) where Fred is the first to look for the electron. In this case the state of the electron when Fred makes his measurement is:"
Here, Fred is outside the light cone of Elle, and therefore simultaneity of these two opening box events cannot be established in SR. That is, event E cannot be said to be before event F and event F cannot be said to be before event E. However, quantum mechanics is a time-dependent theory, and in order to treat the quantum probabilities of entangled systems properly we have to consider the case in which Fred [in Frame(F)] opens his box F before Elle opens box E. If we assume the quantum probability as only the possibility of Elle opening her box first, then SR, I suppose (correct me where I'm wrong), says that we assumed incorrectly that event E occurs before event F.
Ask yourself these questions: If Fred traveled .5 light minutes away would there be a frame where Fred looked in the box first? Why is there a frame where Fred looks in the box first just because he traveled more than one light minute away.
There's no magic to 1 light minute. The purpose of phrasing the thought experiment in terms of one light minute versus one light second (or in terms of distance such as 1 mile, etc), I think, is to show that frame F cannot possibly be the same reference frame as frame E and that occurrence of event E (on earth) cannot be stated in SR as occurring before event F. Besides, it's much easier to say 1 light minute.
I believe that both models of relativity and QM are falsifiable. So far we havn't devised an experiment which has falsified either in there domains of applicability. QM and special relativity are married for quantum field theory [the standard model]. Its not amazing that this is true since the spacetime they are both formulated on is flat.
Quantum field theories are relativistic, but that doesn't mean that all philosophical issues for realism have been eliminated. Even Dirac was displeased with QFT in terms of resolving certain realist approaches to the theory (e.g., renormalization which he criticized as not being a real solution to the divergence problems of field theory). What this is saying, in my opinion, is that there is no satisfactory realist response to merging QM and SR. It says nothing against the theories themselves.