Both Bournemouth and Solent Radar are very helpful at providing FIS in their vicinity. I regularly track low level along the coast just outside both of the Zones. One centre usually asks you to contact/free call the other as you pass somewhere between Lymington and Beaulieu.
Both can be busy but provided that your RT is up to spec then I have never heard either of them be anything but helpful. They may ask you to wait if they are busy and ask you to remain clear of controlled airspace.
There are often plenty of aircraft in that area, both GA and commercial, and I find that both centres provide an excellent FIS that is nearly as good as a RIS.