From the point of view of the legality, both the directive cited in the EuroGA thread and the poster predate the change to Part-SERA, which uses subtly but importantly different wording.
That said, I frequently cross the City CTR north-south in a twin (and yes, Thames or Heathrow Special are brilliant), but I would be very uncomfortable doing it in a single.
For the OP, AIP GEN 3.3 lists in para 6 at the end some contact numbers. The London Terminal Control Watch Supervisor is your best contact for the ATC aspects of your request, though I would expect the answer "we'll see what we can do at the time, subject to traffic". I imagine that NATS will not offer an opinion on compliance with low flying rules.