An alternative route, and far more scenic for a few extra miles
Plan via CALSHOT VRP (big 600 FT chimney at the south end of Southampton Water) going east of Southampton CTR, call Solent around New Alresford (east of Winchester), you can always ask for something more direct, but expect entry into Bournemouth via Stoney Cross or Hengistbury Head VRPs, most unlikely Bournemouth will let Solent offer you a direct approach, but its worth a try if you would prefer.
Despite the occasional bad press that they get, Solent are OK, if there is no traffic in the way, let will let you go direct. If you at any time feel that you were unfairly treated phone them (02380 627189) your complaint will be noted and investigated.
Bournemouth, well that may be a different story..........