It's not "perfect" but if the twenty Southern 171s are unavailable then surely one option would be for EMR to get the sixteen Northern 170s instead?
Given that the EMR franchise have a lot of 156/158s to give up, all of which would be easily absorbed in the Northern fleet - whereas the 170s are non-standard at Northern - that seems a reasonably neat option (compared to some of the things being suggested on here, which involve giving TOCs trains that they've never operated before).
(I think that there are fifty six carriages at Southern, whereas Northern only have forty eight carriages, admittedly)