You don't say if it's a week day or a weekend.
If you can get above 3407 feet, you can just go over the top of the Boscombe CMATZ. If you route TB->Thruxton->Boscombe, you will remain clear of the danger areas.
If it's a week day, Boscombe will provide a DACS if appropriate, and probably a MATZ penetration, though they may not let you into their ATZ.
I did a similar routing yesterday. Blackbushe->Thruxton->Boscombe Down->westward. I was actually at FL60 once clear of Basingstoke, so didn't have to talk to anyone. But had a nice chat with a chappy in a helicopter operating somewhere close to Boscombe! :-)