Bear in mind the usual problem with off airways flight plans - only the point of departure and the arrival airfield will have them. Nobody en-rte.
So, yes, Bournemouth will get you a clearance that they will negotiate with Solent before departure. They should tell them you're going to CPT next (or at least Solent should ask). After that - expect to explain your further routeing along the way.
However - expect direct CPT. Can't see you getting BIA - SAM 3000' for traffic reasons (Both Southampton and Bournemouth), so I wouldn't even plan it at that level. Higher maybe, lower definitely not.