OP, just to remind you of the exemption:
5)The fourth exception is where—
(a)the vehicle is being used for the purposes of delivering goods to, or collecting goods from, any premises, or is being loaded from or unloaded to any premises,
(b)the delivery, collection, loading or unloading cannot reasonably be carried out in relation to those premises without the vehicle being parked as mentioned in subsection (1),
The driver also has to prove to the satisfaction of the adjudicator(I think it's reasonable to assume that the council would reject) that delivering could not reasonably have been carried out with the vehicle parked elsewhere.
But if this got to adjudication procedural matters would also come into play anyway.