I see Great Britain taking up, in stages, pretty much all of Argentina with one or two exceptions, and I see Argentina in 2012 being as powerful as Canada, probably a little more.
So you're saying that Brazil would still have closer relations with Britain than with France even with a British Argentina next door? And the skirmishes and possible wars would revolve, I suppose around slavery as well as the Christie ship of the early 1860s?
Brazil and British Argentina will be having the tense relations and through BA, with the United Kingdom. I'm thinking it will flare up off-and-on through the 19th century, and also in part depending on how the British take Argentina, how quickly, and what they do with the people already there who speak Spanish. Do they push out all of them, or do some welcome the British as liberators and help them out, thus easing the transition of power from Spain to Britain?
Remember Germany and Britain's relationship - Prussia was their partner during the American Revolution and pretty much during the Napoleonic Wars, but after Empire, Germany kind of stumbled around and made most of the world mad at them, leading to OTL WW1 and WW2 (that's the short version of the last 250 years!!).
It's very possible that, if we keep just about everything else the same - aside from British Argentina, which comes before the Monroe Doctrine, so the US would likely not argue it too much - I could postulate Brazil during its imperial phase will try to start wars with Britain, especially if Spaniards come running north with (exaggerated) tales of British horrors, and/or the British push into Brazil proper (Uruguay and the area encompassed by the Uruguay River). Clashes and outright wars, maybe months to a year or two in length, forcing Brazil to capitulate a few times. This antagonism may set Brazil to the Central Powers and possibly the Axis, or both.
With Britain involved in the South, this might also draw more settlers from Canada to British Argentina, due to climate over the course of the 19th century, making Canadian growth slow in relation, and making BA the more vibrant dominion. A posibility is Oregon Territory could be split more in favor of the US - fewer British up there, fewer defenders for the territory, so the Americans may end up north of 49°...maybe up to 52°, as one timeline on the board I had read.
Now, if you go with a Brazil that has had a history of antagonism during the 19th, and one or maybe two international alliances against British Argentina (and by extension, the UK), this somewhat mirrors Germany, which today is not considered an enemy of the United Kingdom, but an ally (at least as far as I know). Depending on how Brazil de-militarizes after your possible one or two world wars, and "repents" and turns away from its antagonism, it could have a new government installed by the Allies that turns the country towards the British and towards British (and likely also) American values.
Just some ideas for you, so take them as far as you wish.