The purpose of the argument is clear. Dick doesn't want anyone to know dick. The bad faith argument's purpose is to shield his papers from public scrutiny via the Presidential Records Act, which provides for public access to executive branch records with certain reasonable restrictions for national security and the like. The basis for his argument, however, is only clear in its asininity.
Dick maintains that because of his constitutional role as President of the Senate, he is actually an appendage of the legislative branch. Granted, he is President of the Senate (a largely ceremonial position) and the Senate does pay his salary and that of most of his staff. And he does get to vote in the Senate when there's a tie. See U.S. Const. Art. 1, cl. 4 & 5. But why, if as a matter of originalism the framers thought of the Vice President as a creature of the legislature and not the executive, did they provide for his election alongside the President and not the Senate? Why, originally, did the winner of the most electoral votes become President and the runner up become Vice President? See U.S. Const. Art. II. Is Dick maintaining that originally Presidential candidates were running for an executive branch position with a legislative branch position as a consolation prize? Does Dick seriously think the framers and ratifiers thought of the Vice President as a special legislative officer who just happened to be elected in the same way as the President and serve the same term as the President? No, he doesn't, he's just doing anything he can to avoid public scrutiny of his public office. Being President of the Senate no more removes the Vice President from the executive branch than the power of the President to veto legislation makes the President a legislator.