How could one of the major organizations within the federal government most effectively overcome the supreme court ruling to allow the federal government to outlaw flag burning

Congress could propose a constitutional amendment that would outlaw flag burning.

Explanation:

The US Supreme Court has decided twice about flag desecration. In the case of Texas v. Johnson and United States v. Eichman the Court decided that flag burning is protected by the right of free speech. In both cases, the US Supreme Court overturned Acts passed by the US Congress.

The only way to enact flag protection statues is through a constitutional amendment. The theory of an unconstitutional constitutional amendment is not majoritarian accepted in the US so an amendment of this matter would be accepted and not overruled by the US Supreme Court.