Well, if we go for the rule of cool then yeah, lets have an Inca king.
Going back to reality, you confused 40 years of Argentinean history.
Let's reconsider what you said slowly.
First, Buenos Aires started the Revolution, it was certainly the head, because it was the most liberal and aware of new ideas city.
Second, Buenos Aires separated from the country in the 50s, 30 years after the last proposal for a king.
Third, it wasn't enormously popular in the rest of the country. It may have been popular in certain areas of Bolivia, but let's face it, by 1816 the Elite of the country was already resigning to the idea that Bolivia could be made part of the county. They still claimed some of South Bolivia, but that's not full of Natives.
Fourth, read what I posted before about racial relations in Colonial Argentina.
Fifth, as I say by 1820, the idea of a king was dead, but I invite you to provide a source that it even was discused as an option in the Constitutional Congress of 1852.
Sixth, how are you supposed to have an Argentinean working as a country without Buenos Aires? Is like having the USA having the whole Louisiana Purchase minus New Orleans. You can have it happen but economic development will be halted and sooner or later the city will become part of the country. And then you are screwed, the economic might of Buenos Aires is unbeatable in Argentina.
I am aware the Inca plan wasn't discussed in 1852, but it was discussed in 1816, at the Congress of Tucuman, which had nearly pulled a fait accompli on Buenos Aires, until Buenos Aires got the whole thing moved to Buenos Aires in March of 1817. My PoD is simply what if the Congress of Tucuman hadn't moved? I think they would have gone ahead with the Inca plan. Also, Charles II, Duke of Parma (then Duke of Lucca) was still being proposed, I think, as an Argentinian monarch as late as the period of the Argentina-Brazil War of 1825-1828. You're right that after the 1820s Argentine monarchism was apparently dead. But the Inca plan (and many other possibilities) were at one point very close to fruition.
http://en.wikipedia.org/wiki/Inca_plan