If the school has decided to take the King’s money, they have to dance to the tune he calls. Decline the $$$ and do as you please.
Note that the Supreme Court has historically clarified the following. Congress is prohibited from laying taxes in the name of state power issues, essentially any issue which Congress cannot justify under its constitutional Article I, Section 8-limited powers, deciding policy for intrastate school lunches not one of those powers.
“Congress is not empowered to tax for those purposes which are within the exclusive province of the States.” —Justice John Marshall, Gibbons v. Ogden, 1824.
So the federal funding that the corrupt feds are now dangling as a carrot to get intrastate schools to accept the FLOTUS’s diet plan are arguably state revenues which Congress wrongly stole from the states in the form of constitutionally indefensible federal taxes.