Only the first sentence is relevant; there is no legal requirement on an individual to provide their name. Period. Everything else in the extract is TVL/BBC's wish , and therefore of no consequence.
So long as a letter withdrawing access is seen to come from the address in question, and that it responds to a letter sent to that address by TVL/BBC, those facts combined prove that the writer is the resident. If TVL/BBC persists, write back:
"As stated previously, your implied right of access is withdrawn. Your letter will be placed before the court as evidence you have received my instruction withdrawing implied rights of access. You state that you will still attempt to access my property - this letter places you on notice that I regard any approach as harrassment".
Photocopy your letter before posting, and send it by recorded delivery.
William Pitt the Elder, 1708-1778