This bill is not
technically unconstitutional with regard to the Federal Constitution's dual-office holding provisions, since it only clarifies who the governor can nominate- not who can actually fill these positions.
However, anyone who's already holding office must certainly resign from their old job if they want to accept the nomination to this regional cabinet; see Article V, Section 1, Clause 1 of the Constitution. Even these advisory positions would be considered an "office of the Republic of Atlasia at any level of government."
Also, I agree with Cincinnatus that the first section of this bill should be reworded; I don't think anyone wants Northeasterners being shanghaied into cabinet service