CAS crossing
I'm a slow aircraft pilot (the plane, not me, hopefully) and a retired ATCO. I have requested CAS crosses frequently and been refused only a couple of times over the years. However I never ask if the controller is obviously very busy and I then keep well clear of the airspace. If you can't get a word in edgeways then stay clear.
Solent are excellent, I live within the Southampton Zone and sometimes like to annoy my neighbours, I don't think I've ever been refused entry or transit. Brize have refused me in the past, but I've now got a mode S transponder which eases the situation. I also try to make the RT accurate, succinct and professional. Never had problems with Thames, Birmingham, Farnborough or EM. Getting and following ATC clearances is part of staying up to speed in the air and keeps you sharp. I don't think controllers intentionally refuse GA clearances as a matter of course. Many ATCOs, especially the older NATS ones are or were active in GA, indeed in the old days before the accountants took over, a PPL was part of the basic training course, so there is a lot of empathy out there.
Would be nice to have a few procedures like the LAX ones - popping out of the airspace just S of Imperial Hwy and turning hard left for a westerly landing at Hawthorn parallel to LAX 25L arrivals seemingly just off your wingtip is a never to be forgotten experience - but the rules and CAS in the UK are very different. Also the IFR routes into and out of LAX have the benefit of relatively much larger airspace volumes which helps to naturally segregate traffic.