Sorry, but an administration charge will embrace costs under a typical section 146 clause which includes "preparation or consideration of", or similar wording under (c) and (d).
You will see LVTs have been variable in whether they were in themselves , as say a service charge is, reasonably incurred, or not.
it's frustrating to say "they're wrong and they also charge me?!" but you have to treat it as fighting a war on two fronts, not trade them off.
Based on the information posted, I offer my thoughts.Any action you then take is your liability. While commending individual effort, there is no substitute for a thorough review of documents and facts by paid for professional advisers.