With reference to ICising former Network SouthEast services, there's a good reason BR kept NSE separate from other areas - the classic fast/slow pattern is much less prevalent, but rather NSE tended to do this sort of pattern:
If we have stations A - B - C, you'd have a stopper A-B and then a train that runs fast A-B then stopper to C.
This works really well where you've got one common core destination rather than a spread out pattern, and should not change on the basis of ideology (much as I am a fan of train categories). Germany notably did used to use this pattern in cities with S-Bahnen and they called it StadtExpress, though I think they're all now REs.
The one obvious IC I can think of that would work south of London is the Brighton Express, presently aka Gatwick Express, as that is a true fast service.
Having said that, I can't see a big issue with including this sort of service in a middle "express" category. We don't have the funding mechanism differential Germany has.
I would say IC needs to be a complete service proposition - faster services, 2+2 in Standard, 2+1 in First, seat reservations available and some sort of catering. As a side example I could see Liverpool-Newcastle 2tph fast as proper IC services as well as Manchester-Scotland, but not the rest of TPE which would just be Express.
Weymouth is an oddity which could be IC (if they upgraded 1st back to 2+1 and put the trolley back) or Express. Salisbury I'd have as Express even though it's a stopper on the outer reaches.