Peter - an interesting question, it got me thinking. First one thing: As I see it, based on how several leading physicists describe the situation, in entanglement there is FTL information being sent - however it is, so to speak, XOR'ed with a random number and therefore not readable by looking at only one side, without additional classical communication. In quantum teleportation, quantum encryption and quantum computers, this is being used and technical applications are being developed based on it, however they require additional classical communication.
I'm not sure about the details of the GHZ scenario regarding your question, I wish there was some kind of software simulation module which one could use in a building-block fashion to configure different scenarios and play with them. However such a simulation system would have to be quite sophisticated to be useful for more than one purpose.
There is an "advanced" technique that might come even closer to what you are trying to do: 'entanglement swapping' , which I understand only superficially. Yet here some thoughts based on what I understand so far:
For your purposes, two pairs of entangled pairs would be produced (A1,B1) and (A2,B2). Then, A can use a Bell measurement on A1 and A2, which if successful, will also entangle B1 and B2 (which is called entanglement swapping). Unfortunately, there are limiting factors, and the question is whether they could be overcome (I guess it is unlikely since clever people have already tried all kinds of things). Yet to a certain extent, B can then test whether B1 and B2 are entangled by seeing whether they are correlated. Of course, they could be correlated by coincidence, so A and B would have to use many double-pairs so as to get a statistical result. But one of the limiting factors is that they can be entangled in different ways, and so these different ways will again interfere with getting a clear result by looking at only B's side, I'm afraid. However, such scenarios make it more and more intuitive, that actions at A do effect the states at B, I think.
Yet your idea is to start with a state that is known to be entangled, and then either disentangling it or not. So one could think about, in the above scenario, whether A could tell B ahead of time which Bell measurements have been successful (and perhaps in which way), and then, when the time comes to send a message, to disentangle them or not. Here I don't know what options are available for doing so, however this sounds like one of the more promising ideas. Of course, there seems to be some underlying principle that makes all such attempts run into some kind of difficulty. But as long as that principle isn't clearly formulated other than through pre-quantum relativity theory, one might as well keep trying... :) ...and, who knows, perhaps there are other features of our universe yet undiscovered. After all, even Einstein was convinced that there would be no 'real' non-locality and/or non-realism at all, not even the kind that has been proven now, even though he was one of those who discovered that it was implied in quantum theory.
As far as I understand, it was discovered only as recently as in 2001, that the influence of random values in entanglement can be alleviated such that entanglement becomes usable for building quantum computers. Personally, my intuition is that there should be a way, given that there is entanglement at all, even though the current state of art in entanglement gives little hope for a FTL communication that doesn't require an additional classical communication.