The runway width isn't really an issue - LCY is 30 metres wide and there are many airports through the Greek islands like JMK, JSI, JTR which are 30m wide and comfortably handling large aircraft like 737s, A320s and 757s. The main thing you have to watch is the reaction time in the event of an engine failure at/close to V1 is very short to keep the aircraft on the runway with asymmetric thrust. Some operators impose lower crosswind limits for narrow runways with this being a key reason. 37m width at SOU imposes no limitations for narrowbody operations.
A combination of length and performance-limiting obstacles are the main problems at SOU. The E145 cannot take a full load and becomes seriously limiting on a wet runway even on arrival. The ATR72 also has limits on departure off 02 under certain conditions. The deforestation (read "tree felling" to normal folk) exercise on Marlhill Copse removes a lot of the obstacles and will improve the limits for those operators impacted by them today. With the extension added, I'd be very surprised if it didn't make operations with the likes of the 738 and 320 possible with a full payload and a decent Med-type fuel load - you might perhaps struggle to get a 738 off to the Canaries, Sharm or Cyprus with a full load and the extra fuel, but I think that's a long way off in the distance!. Quite a lot of reasonably successful airports have performance limitations - departures off 32 at LBA being an obvious one that springs to mind, and getting off 03 at ACE is another one. Although everyone would prefer no limitations, you've only got to reduce the issues to a point where commercial operations are viable, and not necessarily remove every last limitation of any description.