It’s an interesting question. Meese argues in his brief that “there is no statute establishing the Office of Special Counsel in DOJ,” but the OSC was in fact established by the Civil Service Reform Act of 1978 as part of the Merit Systems Protection Board. There are also several other statutes that acknowledge its existence and either grant authority to or impose regulations on it. While Meese does not directly engage with that fact, other parts of his brief argue that the real problem is the scope of Smith’s authority. But Smith’s particular role was created according to Title 28 of the Code of Federal Regulations, the creation of which is in turn authorized by the Administrative Procedures Act of 1946.
Meese does not engage with the CFR either, but he does engage with the section of the US Code that deals with the Justice Department’s appointment of special attorneys (which, confusingly, comes under Title 28 of that code as well). It seems that his argument must be that the rules in Title 28 of the CFR exceed the authority given by Title 28 of the US Code, and that the US Code clearly holds supremacy over the CFR. It is undoubtedly true that the US Code takes precedent over the CFR in the case of a direct conflict, which means that a lot is going to come down to how the Supreme Court reads the Administrative Procedures Act.
So what are the politics here? For one, Meese has disliked the entire notion of special prosecutors, independent counsels, and special counsels ever since he was investigated by one in the late 80s and had to resign from his position as Reagan’s Attorney General. For another, the Administrative Procedures Act is a common target of conservative lawyers and overturning it is generally regarded by conservatives as being they key to dismantling the administrative state. That the APA could be used to grant Smith such broad authority might seem like an ideal entryway into arguing that the act goes too far to be considered constitutional.
But there’s another possible angle here. Meese is a Reagan conservative and decidedly not a Trump conservative. If Jack Smith’s investigation of Trump is unconstitutional, then so are Robert Hur’s investigation of Joe Biden and David Weiss’ investigation of Hunter Biden. So if Meese is trying to hurt Trump, killing two Biden-related investigations in exchange for one Trump investigation (which very well might be restarted and in any case is only one of many investigations into Trump) might seem like a good deal. And if all three of them were dismissed, it would be up to Merrick Garland to determine which of them to restart. Meese might expect him to only restart the Trump investigation (though I doubt that since he initiated Hur’s investigation himself without notifying the White House in advance).
As for whether or not the investigation would have to start at square one: procedurally speaking, yes it would; but nothing would stop the new investigator from starting with a deposition of Smith in order to greatly speed up the process. And knowing where to look is sometimes half the battle.