I haven't seen any discussion on this and so I am not clear on whether it can only add restrictions on top of existing state laws instead of removing restrictions.
Q: Could the US fed gov reduce a state's minimum wage?
Possibly. 1 Minimum wages are regulated under the Fair Labor Standards Act (FLSA).
The Fair Labor Standards Act (FLSA) establishes minimum wage, overtime pay, recordkeeping, and youth employment standards affecting employees in the private sector and in Federal, State, and local governments.
Something as innocuous as purchasing fast food in one state and driving to another state before eating is interstate commerce, but even that is not required.
If Congress decides that a cap on the minimum wage is necessary to achieve a "legitimate end", then the Court could 2 side with Congress. This despite the apparent meaning of "substantial" with respect to "effect on interstate commerce" (see below).
Such legitimate ends could include maintaining more entry-level jobs or preventing a rise in unemployment within the affected states; either or both of which would affect the national economy. (See, What is the case against raising the minimum wage? for its general effects.)
However, such a law would not pass Congress because no Democrat would support it and most Republicans would be concerned about diminishing their chances for re-election. 3
... the three broad categories of activity that Congress may regulate or protect under its commerce power: (1) the use of the channels of interstate commerce; (2) the use of instrumentalities of interstate commerce; or (3) activities that substantially affect interstate commerce.
Neither 1 nor 2 apply. The question then is whether wages paid at any state's high minimum wage substantially affect interstate commerce.
In United States v. Lopez, the Court identified activities having a substantial relation to interstate commerce as being subject to Congress’s Commerce Clause power. Consequently, Congress’s power extends beyond transactions or actions that involve crossing state or national boundaries to activities that, though local in nature, sufficiently affect commerce. The Court has stated that, even activity that is purely intrastate in character may be regulated by Congress, where the activity, combined with like conduct by others similarly situated, affects commerce among the States or with foreign nations. This power derives from the Commerce Clause supplemented by the Necessary and Proper Clause.
... the Court stated, the commerce power ‘extends to those activities intrastate which so affect interstate commerce, or the exertion of the power of Congress over it, as to make regulation of them appropriate means to the attainment of a legitimate end, the effective execution of the granted power to regulate interstate commerce.’ Judicial review is narrow. A court must defer to Congress’s determination of an effect if it is rational, and Congress must have acted reasonably in choosing the means.
1 I am only willing to stick out my neck just so far. There is no suitable precedent I could find.
2 Still preserving my neck.
3 My neck is safe.