The problem is that letters of marque and reprisal only work if all parties agree to them. The reason that American privateers with Letters of Marque preyed on British shipping during the War of 1812 was because British privateers with Letters of Marque also preyed on American ships. Americans recognized British Letters of Marque and the British recognized the American letters. Any ship captured without a Letter of Marque was considered to be a pirate.
The reason that Letters of Marque don't work today is that as an instrument of international law they were abolished in the 1850s by the Treaty of Paris which also ended the Crimean War. The US was not signatory to the treaty but used it during the Civil War and the Spanish-American War to act against privateers sailing for the Confederacy and for Spain.
Bottom line, since about 1856 Letters of Marque and Reprisal are worthless pieces of paper. They have zero standing in the international community. In this case Dr. Paul is about 150 years behind the times.