I have never heard of a British invasion of Peru OTL...
Their invasions of the River Plate during the Napoleonic War were planned as a grabbing a base from which to invade Peru, I believe.
There seems to be an assumption that the UK can just walk into Argentina, but as OTL showed it would face strong local opposition.
This is a good challenge, and one I'd like to explore more. I guess the arguments for why it would be more likely to be successful in this timeline would be (a) greater focus on the Americas from the British, and thus more resources committed to it (b) more manpower from America, (c) less of a national awakening in the Americas due to no influence from the American Revolution, and, possibly (d) an earlier attack due to no ARW setting the UK back.
And it's not like British North America needs more farmland. So why bother taking it?
For me, this is much clearer. It's a base from which to penetrate trade in South America, in the manner that Hong Kong was for China. In addition, the Cape is one of the choke points of world trade that makes a strategic control point for the British.
If Portugal doesn't move its capital to Rio de Janeiro, then Brazil will fragment on independence, no matter when it is.
Along what lines do you think Brazil would fragment? When would independence occur? Would the successor states likely be British aligned or not?