FAA inspectors acting under the authority of the Administrator carry that authority in their duties. Bear in mind that the duty issued the Administrator is under an Act of Congress (as amended). Each representative carries the full extent of the authority delegated under the Administrator.
Don't expect the FAA Administrator to show up on your ramp, and inspect you. Or correspond with you. That's why FSDO's are set up throughout the country with inspectors and personnel to represent the Administrator.
An Inspector does not carry all the duties and responsibilities of the Administrator, but he or she does carry the full authority of the Administrator insofar as the duties delegated and assigned.
Among the duties not assigned is interpretation of regulation; that's assigned through the Regional and Chief legal counsel's offices. That said, the Inspector carries a lot of latitude in initiating enforcement action and application of the regulation. Until their word is challenged under an ALJ, the word is the law. That is to say, much like enforcement action, the FAA inspector sets the rule at the local level, and if one doesn't like it one can kick it up the chain...but it's always a matter of guilty until proven innocent, and it's up to you to do the proving by appealing and then making the case. If the FAA is simply administering and not taking enforcement action, there's no particular appeal process, although you can go to region to seek assistance.
Use some care in the politics of how the matter is handled, as you still need to deal with the local FSDO.