That any action attempted would be rooted in the Commerce Clause, the line being that state restrictions interfere with interstate commerce. This has been read very broadly since the New Deal, and pressing against state restrictions might have precedential support. It is important to remember these people never stop this side of the grave, and this could be seen by Federalist Society types as a sort of win/win affair. While overturning the Governors would be a vindication of the cheap thug's dictates, upholding the Governors might provide precedents for restricting wide application of the Commerce Clause, which in future could provide support for future efforts to restrict the reach of Federal law and regulation, on which many things, like work-place safety and even a Federal minimum wage rely.
Since the Constitutional grant of power is to Congress specifically, it is not clear to me the Executive has standing to bring the action. Some specific law would have to be found the Governors could be held to be acting contrary to, which an action could vindicate. Or perhaps a bloc of senators might be the plaintiffs. In Wisconsin legislators are bringing suit in state court against the Governor, citing state law. Where there are Republican legislatures and Democratic Governors, suits might be brought similarly, and attempts made to get the matter into Federal courts, with Barr enlisted in aid should that occur.
It is not a totally toothless threat, though I do not expect much to come of it. Courts will be likely to dawdle, and corpses will pile up where restrictions are being relaxed in sufficient number to put the lie to claims the Governors have no legitimate basis for exercising state powers of public health and policing.
"Defeat of a hated enemy is something to be for."
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