According to CAA O&D data lots of pax (almost 50%) originating from the entire SOU catchment leak to LGW where easyJet has a significant presence. I estimate that around 3-5% of all LGW pax originate in the SOU catchment area of which a considerable amount are regular travellers. From this data alone I would assume easyJet wouldn't want to dilute their LGW operation by setting something up at SOU that could cannibalise their pax
SOU hardly sees any leakage to BRS airport hence why the two bases at LGW and BRS compliment each other. If easyJet sets up a LHR operation as they have stated before when the 3rd runway becomes live then SOU becomes even more irrelevant
For clues as to easyJet's strategy or behaviour just look at the fact that they don't operate from BHX for this very reason (leakage to LTN, BRS etc)
Jet2 however would be a safer step for SOU - in my opinion a much more suitable market segment that SOU is *not currently tapping* and Jet2 have no operation at LGW. Plus it would be very complimentary to SOU's Flybe market. An easyJet operation may not be so complimentary (BRS ops have considerable UK domestic from there EDI, GLA, NCL, BFS, INV, ABZ)