Louis XVI dies early, in the late 1780s. The regency of his son is controlled by his very reactionary brothers, who with foreign support come down hard on the revolutionaries, starting their own kind of reign of terror. Many Liberals flee, most to Britain or other nations, but some to Louisiana. This provides a boost to the Francophone population there, and Napoleon gets unlucky and has to flee as well.
The United States, seeking to expand its influence in the face of a powerful Spanish Empire, works closely with Francophones to create a Republic of Louisiane that's aligned with them and will cede most of the northern Louisiana Territory. Napoleon due to his talents quickly becomes a key leader of this conspiracy, and When Spain is distracted in another war it succeeds.
The Republic of Louisiane is Democratic in name, but is more or less controlled by Napoleon and the military, who are themselves heavily influenced by the United States. Napoleon is elected virtually unopposed to multiple terms as President, where he leads the nation to shoot well above its weight. He tries to attract as many francophones, particularly liberal ones, to Louisiana as possible to offset the many American settlers who have a fair amount of influence in the nation. Working closely with the U.S, he fights off Spanish and British encroachment in the Mississippi and possibly expands Louisiana into some of Texas, though he has to give up most land north of the OTL Orleans Territory. His quest to hang onto power is only matched by his quest to keep the country independent of U.S control. The country is very liberal, abolishing slavery and giving similar rights as those handed out in the French Revolution.
After Napoleon's death, Louisiane develops as a middle-sized Republic heavily tied to the United States. Although French remains its language and the language of the people, waves of American, German and Irish immigrants have a huge effect on the culture of the country, particularly outside of the more French New Orleans. Today Louisiana still has a close relationship with its large neighbor, the United States, but has managed to create its own culture and has had an influence on world affairs. Territoriality, it covers OTL Arkansas and Louisiana, plus parts of Texas and Oklahoma it took from Mexico. It has become much more of a true democracy since the early days, but to this day Napoleon Bonaparte is revered as Louisiane's own George Washington, and well respected by military historians as an accomplished general in his own right.