David (et al):
I'd say CAP 419 would be fairly clear on the matter.
Unless you've been cleared to cross controlled airspace, you'd be entering CAS without a clearance. Approval to 'transit' would, to me (and on searching through CAP 419, I find nothing to contradict me) merely mean approval to fly through the overhead -- remaining clear of controlled airspace.
Despite not having JSP552 in front of me I'm pretty sure it's the same.
Of course, the sensible thing would be to check with the controller: a transmission along the lines of 'confirm cleared to enter the class D' would soon remove doubt on both your parts.
I suspect that Brize are significantly less bothered about their ATZ than, say, Lakenheath, who fairly often issue MATZ crossings with the request to remain clear of the ATZ. Brize is home to large transport-type aircraft, flying much larger circuits rendering the protection of the ATZ largely, I suspect, null and void given the Class D CTR in which it resides.
Never assume: it makes an ASS out of U and ME...