How could the rising British attempt to conquer significant chunks of Latin America after 1800? Belize, Mosquito Coast, the Lesser Antilles, etc. don't count.
How successful could they get?
If the states just become puppets, that fulfills the challenge as long as they have a class of British traders, soldiers, or craftsmen living in them. British including Irish if they are supportive of the British state.
They did try, and failed. They invaded Buenos Aires, in 1896 and then in 1807, but were expelled. They had plans of conquering Valparaiso (Chile), Buenos Aires and Montevideo. From there, they might have expanded inland.
But then, in 1808, European alliances shifted and Spain (or what was left of it) become and ally against Napoleon, which ment that expeditions to their empire weren't politically feasible any longer. And then, when the Latin Americans revoltedbelled against Spain, the new rulers of the new republics were willing to trade freely with the British, and abolished all restrictions to trade with foreigners (non Spanish) that had existed during colonial days. So, the british saw now reason to conquer them.
In fact, IOTL, the South Cone was firmly on the British sphere of influence. Banks, Railways, insurance companies, bonds, all that was in british hands. And most manufactured goods came from Britain. There was even a sizable british colony, formed of traders, bankers and railway engeniers, that, while beeing waaaay smaller than the Italian or Spanish communities, kept very close, was very worried about mantainening their identity, and was highly influential. These British, however, usually didn't became big landowners, except in places like Santa Cruz or Tierra del Fuego, where the soil was the poorest.