Unlikely, the place was too far away. Rio de la Plata would likely become a mess of warring caudillo statelets.
As for the rebels taking North Peru, who's to say that a fair number of the local natives wouldn't rise up on their own after hearing the news that Cusco and La Paz fell? While Lima would obviously never fall to such an internal uprising (the city would have to be taken from the outside), maybe a place such as, say, Cajamarca, could.
Who is to say that there would be a united Argentina though?
OTL it was the northern provinces which supported the idea of making an Inca King, and Republican Buenos Aires which opposed it. I could see Buenos Aires going its own way but the Congress of Tucuman opting to join Cuzco. Given that the Tucuman Congress proposed Cuzco as its capital, I don't think it'd be an implausible thing for northern Argentina to prefer tying its fate to Cuzco instead of Buenos Aires.
Paraguay, Entre Rios, Buenos Aires, and Uruguay would do their own thing TTL as they mostly did OTL though, I suppose.
The issue with going up to Cajamarca is that there's more of a geographic barrier. It's one thing to take control of the Altiplano since it's a single big plateau and another to march over to lower Peru. Perhaps Cuzco could conquer the rest of Peru (and Ecuador?) later on though when the Spanish Wars of Independence are raging.
Come to think of it, I wonder if there might be butting of heads between Bolivar's expansionist Colombia and a Kingdom of Cuzco intent on reclaiming Inca lands. Ecuador and North Peru would be a clear source of friction between the two. There was already war between Gran Colombia and Peru over Ecuador OTL.