Chevvron,
Firstly, thank you for all you hard work at Farnborough Radar. I hugely benefited from your service when I worked at Denham a decade and a half ago - I was working the first morning of the change to Basic and Traffic service.
With all due respect, the current edition of CAP413 generally advises on first contact to give callsign of agency, then aircraft callsign, then initial request. On page 24 of Chapter 11 'Phraseology examples' Figure 26:
.Wrayton Information, G-ABCD, request Basic Service
again on the following page Figure 27
Westbury Approach, G-ABCD, request Traffic Service
There are numerous other examples through CAP 413 where the initial call is formatted like this.
You're absolutely right that the pilot shouldn't launch into their complete message (as I had one slightly nervous student do the other day, before I could stop them - the controller handled it with aplomb). However, a short introduction of what you are wanting takes up little air time and gives the controller an expectation of what is to come.
Incidentally, to answer the OP's question, CAP 413 also says the response from the controller should be simply 'pass you message' and the squawk instruction should come AFTER your detailed message, so you shouldn't have to be fiddling with the transponder until you've got your message out.
TOO