From what EB says, if there's a party of 4 to bring back, plus pilot = 5. What about baggage?
Straight away, that rules out most light singles. Now we're talking twins which are a lot more money and much more scarce.
EB - you might want to talk to one of the charter outfits at Blackbushe. They might give you a good rate in return for some marketting, and you'd be almost guaranteed to get there whatever the weather & time of night. You'd almost certainly need to land at Norwich (for runway lights & approach aids). Yoy could consider the other way round & talk to Sterling Aviation at Norwich too, if you're not worried about collecting them yourself.
I'd guess that you'd have to expect something around £400 each way.
I used to drive every week from Farnham to Ipswich, up on a Mon & back on a Thurs or Fri. I sympathise.
Have you looked into commercial operations between Norwich & Southampton?