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Cheers for having a look at the paper!
The intermediate settings in Section 4.2 are supposed to be the alternate setting from whatever the final setting is set at, in that run of the experiment (and so other runs are not relevant to the calculation). This then gives probabilities for the actual measurement pairs matching EPRB probabilities. (Since the intermediate measurements are summed over, it’s not like they’re actually measured.)
Theoretically, they’re chosen as those settings simply because, as a result, the amplitudes cancel to give the correct joint probabilities. More practically (this is the original idea that led Gerard to think of the model), under the right circumstances, one could tune the intermediate measurement to be (i) no measurement at all (normal EPRB correlations), (ii) a projective measurement (destroying the correlations) or (iii) a partial measurement with correlations somewhere between these extremes. This would then allow an experimenter to tune between quantum and classical phenomena.
On your more general point, yes—this is exactly the sort of thing I’m thinking following on from this. I think there’s scope to point out some simple SEPRB-type cases that involve fine-tuning (one very simple system features in a pre-print of Huw and Ken’s that Huw kindly showed me) and then argue that they are, rather counter-intuitively, subject to Wood and Spekkens’ analysis too.