Utter wrip-off nonsense, in my opinion.
I'd be tempted to go back to the inspecting company and ask them to put in writing the inspector's justification for a C2 coding when after a discussion with the NICEIC you've been advised that should only be a C3.... bet you get an amended report out of it.
We have looked up the building regs now as well and worked out they have misquoted them slightly. They have said british standard 421.1.7...no provision of AFDDs on circuits supplying sockets and applicances. The british standard only quotes "AC final circuits supplying socket-outlets...". They have added "appliances" on, which the british standard does not mention. We therefore think their quote of 5x AFDDs is excessive and calculate we think we only need 2. I explictly asked the quotation to be the minimum to bring it up to standard so am not happy about that. I have done exactly as you suggested and quoted what NIC told us to them to see what their response is