So these letters are specifically NOT mentioned in, or authorized by, our founding documents?
Of course they are.
(1) However, the Constitution of the United States cannot compel any other nation to recognize any letters of marque and reprisal issued by the Congress of the United States.
And, as a point of their own domestic laws, no other nation on earth recognizes letters of marque and reprisal issued by the Congress of the United States or any other country.
(2) The Constitution places no obligation on the United States to recognize letters of marque and reprisal issued by any other nation.
And, as a point of our own domestic law, we do not recognize such letters issued by foreign governments as having the force of law.
Congress is authorized by the Constitution to issue letters of marque and reprisal all the livelong day if it desires - but no one outside the United States will recognize those letters as having legal force. It would a completely pointless exercise.