Mexican Lousiana probably is unsustainable, but I don't know about British Louisiana. Why not just move to British Louisiana and become British citizens? The folks are white and speak english, and until 1833 they'll even let you keep your slaves. The notion that "US will unite to take Louisiana from the British because they want it to be AMERICAN" seems anachronistic. Remember, at the time people were more Virginians, New Yorkers, Pensylvanians, etc. than they were Americans: Texas was originally independent and might have stayed that way if it weren't for the fact that their local government couldn't afford a pot to piss into.
It sounds nice, but it is unlikely to work. My ancestors were in this group, some were in Arkansas pre-Louisiana purchase. They had left Europe by 1750, perhaps by 1700 and did not think of themselves as English. So in many ways, this is asking why the Czechs don't just decide they are Austrian. Identity is a lot more than just language. Perhaps I should say it is like the Prussians just deciding they would rather be Austrians.
Also, while I am consider white today, many of these people are not-white by European standards, much less Anglo-Saxon. For example, my Dads side is Basque/Laplander. For other you have a lot of Native American blood or even escaped slave blood (greatly diluted from African). They would be trading largely independent villages where they can vote for being UK colonials. It is not in their best interest.
Now sure the UK could possibly prevent the USA from getting the Louisiana purchase, but it would require it being settled with non-Americans. And you are right, more non-Virginians or in my case non-Carolinians. The only population stream I see that is both big enough and loyal enough to the UK is the stream that went to Australia. So I would say a TL where France traded Louisiana for Australia is the only easily plausible one. Now if we want to do a Mexico analysis, we need the same thing, Mexicans to move to Louisiana in large numbers, or at least Spaniards. But Spain has so much other land, why choose Louisiana? Why chose a Malaria/Yellow Fever Swamp over say Argentina, Costa Rican highlands, or central Mexico. White people (generally lacking genetic adaptations) to almost always avoid Malaria swamps. Now one can right a TL where they go straight to Iowa, bu that requires some very complicated logistics compared to anything France, England or Spain ever did. It would require first putting a decent size army (10,000 soldiers) or so directly in Iowa and bringing up supplies up/down Louisiana.
And the USA had an easily logistical situation, all no fault of the USA government. My family makes a good example. They were west of British control by 1750. They would move as a group west, often about half a village at a time. They stayed in the same type land, far from disease that they new how to live on the land. They brought their own military power and combat experience. They had long history of working with/against various Indian groups. And this shows what any European power would have needed to settle. You don't need individuals to move over, you need entire villages complete with militias to move over.
And France had a side issue of declining birth rates so lack of population to export.
