Approximate synchronization and the arbitrary horizon is fine with me. What isn't is the assumption [A] that on either side of the horizon an inertial frame exists where we can conclude that either Elle or Fred made the measurement first.
I didn't perceive the same assumption from this thought experiment. What I perceived is that special relativity requires that we accept the possibility that, ontologically speaking, Fred might have opened the box before Elle. That is, special relativity is agnostic about the chronological order. It is this agnosticism which forces us to include 3 different scenarios using Born's rule.
Relativity can only speculate about unobservable events [such as what happens to observers falling into a black hole, etc.]. For this thought experiment it is easy to show that [A] isn't correct just by allowing more time for information to arrive describing Fred's actions.
Would your statement that relativity can only speculate be the same as saying that relativity is agnostistic? If so, then wouldn't you have to include all the possibilities in your quantum equations which force the logical inconsistency between both theories? But, [not-A] is based on the assumption of what we could conceivably show this within the context of SR, however in Fred and Elle's case, we can send signals that would give good indication of re-establishing simultaneity, but since they are no longer in each others reference frame (i.e., Fred has moved outside the inertial frame with Elle), we must, I think, remain agnostic from the perspective of special relativity about the exact simultaneity of these events. The only exception is if you are saying that Elle and Fred are in the same reference frame at 1 light minute away, but I don't see how that is possible in SR to say that.
How about having Fred fall into a black hole [where a natural horizon exists] and then having Elle and Fred make their measurements. It would be easy to speculate what relativity would conclude using a proper frame metric.
If Fred travelled beyond the event horizon of a blackhole and made the measurement, he would be beyond the light cone of Elle (even if 2 meters away), and again SR, I think, would be agnostic about which measurement happened first. Sure, we can speculate, but SR doesn't give us any means to establish whether two events are synchronous if they are different reference frames.
Keep in mind that relativity doesn't have anything to say about the non-local activity. It would only have something to say about the time ordering of the events and where they took place.
My understanding might be incorrect, but wouldn't SR be agnostic about sequence of events that are outside each other's reference frames?