As stated above, Solent is not a LARS unit as such....
Just because a controller does not sound too busy on the R/T, it does not mean they are not busy with everything else they have to do.
Solent have had a huge increase in traffic and it is set to continue... In my dealings with them as a TC controller, I think they do an excellent job.
As for being anti GA - I doubt it, but they do not get paid to provide a LARS service by the Governement, so if they refuse for a valid (or even non valid) reason, then thats up to them, and good luck to them.
I am not anti GA myself or anti training flights, but I would love to ban A/C that can only climb at 300 - 500 fpm and do 120kts from my airspace... they pau less money and cause ten times more hassle than the airlines, because they are always at awkward levels and in the airspace for such a long time.
As a fairly new PPL, you should try to get yourself a visit to Bournemouth Tower, Solent radar, and even maybe TC... then you will see the sort of interaction that goes on, and the workload that can be present, even if it sounds quiet.
Enjoy your flying and remember, a refusal is usually for a good reason, whether is is obvious to you or not.... your transit may coincide with the planned departure of several A/C that you do not know about!!