Could Burr have made himself a king of his own nation?
Part of the problem with this question is that Burr himself never really decided on what he was going to do exactly. Was he after New Orleans? Parts of Texas? All of Mexico? Washington DC? No one knows and it's pretty damn likely that Burr never knew himself.
He kept things rather vague and nebulous throughout the whole life of the "plot". He seems to have thought this allowed him to keep his options open but he made so many different promises to so many different people that there is no way he could have kept a fraction of them. The best guesses we can make nearly two centuries later about his thinking was that it was an early filibustering expedition: He'd set up a government of some sort within whatever he was able to buy/grab and then apply to the US for statehood/territorial status. I've read one account which suggests Burr thought such a landgrab would restore his political fortunes within the US, i.e. "vote for me because I added X, Y, an Z to the Union" or "March on DC with me because I added X, Y, and Z to the Union
Whatever he meant to do, the European powers involved, like the UK or Spain, could have smashed up anything he managed to put together. Even the US would have likely made an attempt to put Burr down and not just because his actions as a US citizen seizing territories belonging to European nations during the middle period of the Napoleonic Wars would have put the US at great risk.
Burr probably could have achieved a short lived success in the region, but his days would have been numbered.