NEC3 ECC: Should Contractor get fee if Employer pays costs?

The Employer has paid for costs that we as a Contractor were required to pay for charges from a local authority. Should any credit be provided and include the fee percentage?

Our view is that they fee should not be applied as we have done the works included in the fee this is only a mechanism for reimbursing costs the Employer has paid as this is who the authority charged.

I am not sure what option you are under here. If you are under options A-D, I assume you have included those charges in your price - so why would the Employer have paid them? It would then need a CE to lower the total of the Prices? If an Employer omits work then yes that would be a compensation event and any reduction would include fee so you do not get to keep it.
If you are under option E, then they are not obliged to give you any amount of work or cost so you would not be entitled to fee either here.