my imagination at work ....
I don't know why Louisiana it self is so thin, but that is is a beautiful map!
As much as I love anything Francophone, this is a tough one. But, I think it can happen if France does some specific things with their population. For example, establish a penal colony in the western coast and attempt to have them move inland as much as possible. Throw some Protestant French/Germans there too, along with some estranged royals and the like. Then, try to have as many Catholic immigrants to flood there from places like Italy, Portugal, Ireland, Poland, etc. That should give you a good population for the basics. Then, just get Napoleon or someone else to create a "non-colonial Sister Republic" or something similar there to show how "free it is". The US would still be interested in expanding west, but for the time being the Louisianan Republic could prove to be a useful ally or buffer against Spain.
This sounds pretty good, perhaps have the French establish a penal colony in the middle of nowhere up north, perhaps in modern day Kansas? Also have the French encourage Huguenot immigration to Louisiana. Then have Haitian Revolution survivors flee to Louisiana, not only whites but also mulattos, who both bring a significant amount of wealth with them. They settle farther North, especially the Mulattos since I'm sure New Orleans would not be so well treating of people of a mixed racial background. If only I could shift early Irish immigration to Louisiana but I don't really know how to do that. I think if Jefferson decided it was unconstitutional, or instead if Napoleon decides to add the condition of U.S alliance with France, which Jefferson would refuse. Then Napoleon basically has no choice but to let the colony do whatever it wants to, I could see them being trying to attract immigration from Europe during a time of war, which means thousands will come, and after Napoleon is defeated (Perhaps Louisiana helps the U.S during the War of 1812), thousands of refugees come to Louisiana. Eventually it'll be pro-Bonapartist enough to declare an independent state.