At the time of independence (1808- 1810), there was significant sentiment to having a monarchy. Carlotism was mostly a dead end (besides being a woman, Carlota Joaoquim was the wife -hated, admittedly - of the Portuguese King Joao VI, and an absolutist to boot). But it seems as though, IF a king could be found, it was a quite possible scenario. Nevermind the 'my horse for a kingdom' (who would be king) problem, IF a king could be found, how large a country is reasonable to expect?
Buenos Aires is republican sentimented, but could be persuaded if the capital was BA. upper peru (basically modern day peru) and northern Argentina were extremely royalist. I think they'd be on board. Chile? Uruguay, Entre Rios, Paraguay were more 'don't really give a shit one way or the other'. If the king/queen looked happening, I'm guessing they get on board.
How big could it be? How long could it last? Are the regions within a Argentina/Chile/Peru/Paraguay/Uruguay compatible? I know the portenos (BA) are a huge problem to harmony. Paraguay is isolated, so it tends to be 'leave me alone or I'm going to secede'. No idea on Chile. I'm guessing that if you could leave BA out, there's greater chance for harmony, but then you leave out the main port. Montevideo is a better port, but the routes to it are tougher.
Does Brazil embrace a huge monarchy to the south or reject it? I know Joao rejected Carlotism because he didn't trust his bitch of a wife (egads, just think if they worked together!).