I reiterate the usual points myself and others make regarding French Louisiana becoming its own country--not plausible past 1763. New Orleans also has the issue of not being centrally located in addition to the obvious (being flood-prone, being in a swamp, etc.). Where "centrally located" would be depends on the nation's borders, but it's pretty hard to justify New Orleans being centrally located.
But I suppose it isn't totally undoable. For one scenario, let's have the US's early internal politics get more and more unstable (so probably POD sometime during Articles of Confederation era), and the western settlers in Kentucky and Tennessee (probably neither state is formed) and basically the whole Trans-Appalachian region decide to go join Spanish Louisiana in exchange for some basic privileges and rights. They defeat US attempts to reclaim them, and this fosters a sense of nation building.
Meanwhile, Louisiana (the nowadays state) remains largely French-dominated. Spain still turns the place over to France, but this increases internal tension. French fleeing Saint-Domingue flood to New Orleans, and France also attempts a mild colonisation process in Louisiana (state). British invasion of New Orleans further increases tension, but is defeated in the end. After Napoleon's final defeat, a combination of French and Anglos declare their independence as the Republic of Louisiana to prevent their territory from being used as a bargaining chip.
Louisiana (state)'s development occurs differently from the rest of the country. By far the most Francophone part of the country, Anglo settlers still come to Louisiana, but they never become a large enough minority to truly take control from the French. So Louisiana develops a bit like Quebec--a Francophone state in an Anglo nation. By the 20th century, strong movements of identity emerge in Louisiana, which make the state officially Francophone. However, the Anglo minority is politically strong enough to ensure that places of major Anglo settlement (like New Orleans, the bordering regions between states) retain a degree of bilingualism.
With an earlier shutdown of the Trans-Atlantic slave trade and even better enforcement by all parties, the supply of new slaves was constrained early on. This drives the price of slaves up immensely in Louisiana. Following a slave revolt incited by American slaves shipped across the border, Louisiana banned the slave trade between it and the United States. When the profitability of slavery is weakened as well, the practice dies a slow death, and slaves are gradually emancipated--first children born to slaves are freed, then the elderly slaves are freed, then finally in 1879, Louisiana abolishes slavery entirely.
New Orleans remains capital thanks to a gentleman's agreement that "it's the least worst option". Being capital would help it's development, but I'm not entirely sure how to do it. One way might that once slavery is abolished, the inevitable Great Migration to the north is lessened by some civil rights laws--no government-supported segregation, as well as less lynching and less of a climate of violence. In the north Louisiana, there's less of a pull factor going for black labour. During migration to urban areas, New Orleans thus gains much more black migration. Ensuring good race relations would also reduce white flight. Combine that with better economic development of the capital, then you'd get New Orleans a much, much bigger population. Sadly, I don't think a nation founded by Tennesseans and Kentuckians that also includes Mississippi and Alabama is going to ensure good race relations, but I think avoiding a Louisiana Civil War and having the *Louisiana Supreme Court rule against Louisiana's version of Jim Crow might help somehow.
Finding a way to draw off Texas's oil wealth from flowing to the Texas Gulf Coast would help New Orleans out as well, though it could draw it to a different area of the Gulf than Houston (or Galveston, for that matter). Luring the auto industry or others to the New Orleans area (like how the auto industry is all over the South nowadays OTL, but that assumes that Louisiana gets a Rust Belt) would also help, as would keeping oil prices high in ways that don't involve hurricanes striking the New Orleans area.
Spanish language is official in New Mexico, which joined after a revolt against Mexico in the 1850s (Texas and the rest of the Mexican Cession was gained in this time too) and placed themselves under Louisiana's protection to preserve them from both hostile Mexican authorities as well as more importantly, Indian raids. Even more Latino in culture than OTL, New Mexico is officially bilingual. Speaking of language policies, French and English are each official languages in this Louisiana--Canadian bilingualism OTL is similar to what I'd imagine. Spanish is important to learn, thanks to Latin American immigration as well as New Mexico.
The other scenario I had was the classic "Napoleon flees to the New World". From there, he recruits settlers from the US, and a similar scenario above occurs, with the main differences being that the nation is a (constitutional, it would have to be) monarchy under the House of Bonaparte and it lacks the east bank of the Mississippi. Lacking the east bank of the Mississippi will change things bigtime. I think most importantly, the slavery lobby will be a smaller one and it could die an earlier death.
There, I think I met all of your criteria plus some, although I'm still not sure if my idea for New Orleans would get the city to that size (but definitely much bigger than today).