An interesting thought. Southampton has the potential to be an absolute gold mine for an airline like Jet2, providing they can iron out a few major issues.
Now we all know that Southampton has handled 737 800's, A321's and 757's in the past on an ad-hoc basis and A319/20's and E-195's on a regularly scheduled basis. However there are a number of important points.
1. People love Southampton, SOU/LGW have essentially the same catchment outside of London. The main difference is that most passengers prefer to use Southampton if it suits their needs. This relies on Southampton being able to absorb a 4, 6, 8, 10... However many aircraft base in the long term and still keep its attraction as a convenient, friendly, easy to use, small airport with brilliant public transport links. From personal experience, one of the best in the country and I love it. London City has managed to get up to 5mppa and keep its attraction, so it's not impossible. If Southampton clogs itself up though and becomes a sh*tshow... Advantage Gatwick.
2. Runway length. The extension will put SOU between SEN and BRS in terms of runway length. You can launch an A320 to the Canaries out of SEN and pretty much any narrowbody to anywhere meaningful out of SEN. So I'd be optimistic that the 738 SFP at LS could operate a meaningful number of their bread and butter routes, year round pretty reliably with only minimal penalties. I'd hope the same could be said for the A321neo, it appears to have great performance and Thomas Cook managed the bog standard A321 out of Bristol with 212 punters onboard... The A21N has to be better.
I reckon it's safe to say that continental Europe isn't going to be an issue at all. The canaries, further flung Greek islands and Cyprus might take occasional penalties.
3. Is there enough room for SOU to handle (for example, medium term, not an unreasonable estimate) 8 based aircraft up to the size of an A321neo. Park them overnight, boot them all out within 2-3 hours in the morning before receiving, turning and booting again in a wave in the afternoon? Firstly it appears that would take up half of the available stands. Stand 1 can't be used. Stands 13 and 14 are push and tow to A6, then start up, which would be a bit of a ballache but not the end of the world. It appears some work would need to be done to the terminal and apron to make SOU attractive to Jet2 and able to support a sizeable base. I can't see Jet2 being interested if they were capped at 2/3 aircraft.