You really have no grounds other than a plea for discretion as you don't have an exemption. They ought to consider a toilet break for a black cab drover.
Westminster policy is to give 2 mins observation for Hackney carriages (but for boarding/loading) and there is none on the PCN but generally I think they are friendly to black cabs.
There is a taxi rank outside the hotel which I presume was full but is a point.
