Bournemouth's airspace stops at 2000, and Solent is over the top. If I were to transit Bournemouth at < 2000' I'd call them, otherwise I'd call Solent.
For StonyX to Beaulieu <2000 you don't need to talk to anyone, but if I wanted a RIS I'd call Bournemouth. Above this I'd call Solent for a transit.