katchen,
Britain's big mistake was abandoning Uruguay, when they didn't have to. They weren't kicked out. They would have had the muscle to hold that small country. But, when kicked out of Buenos Aires, they freaked and needlessly left Uruguay, too. Within a scant few years, they were advocating a small independent Uruguay so that they could control trade in the Plata region. They could have had that in 1807. Eventually, they could have de facto controlled the region, but not in the manner you suggest.
The rest of your post is mostly wank, though. They need plenty of troops to conquer and hold all that region. Simply occupying Bahia Blanca, and making anything out of it, is going to take significant resources to pacify the natives.
the easiest course of action is to hold Uruguay, control the rivers, and control trade. Eventually, they'd control the region economically. Militarily, they have a tough row to hoe.
Maybe not. When I started really researching the history of the region, state by state and province by province, I discovered that there were quite a few attempts by regional caciques (chiefs) to create breakaway states that the central governments of Brazil and Argentina put down with great difficulty. The two that most interest us are Ramirez's "Republic of Entre Rios" (1820) see Wikipedia "Republic of Entre Rios" and Wikipedia "Entre Rios" History and the "Ragamuffin War" in the 1820s in which following Uruguay's secession from Brazil, what is now Rio Grande de Sul state also seceded, and remained independent for a few years before Emperor Dom Pedro I was able to put it down. And then there is the conflict between Manuel Rosas in Buenos Aires and the "Insular League" centered in Cordoba, that starts in the 1830s and goes 15 years IOTL. Maybe if the British gobble up Entre Rios, Corrientes and then equally dictatorially run Paraguay, this conflict will not happen. But maybe it will.
In both cases, if the British had owned Uruguay, the British would have been sorely tempted to manipulate such a situation to Great Britain's advantage and to incorporate these areas into the Uruguay colony. The British certainly took advantage of similar situations elsewhere in the world.
And the British would have a moral imperative behind them: The elimination of slavery and the extension of free soil. Both Argentina and Brazil (and Paraguay) have legal slavery. The Argentines do not abolish slavery until 1856 IOTL. Brazil, not until 1885 IOTL.
But Uruguay, taken over in 1807, but POD not given back to Spain in 1808, has few slaves and the slave trade is abolished in 1808. So slavery will almost certainly be abolished in the new colony, as it is abolished in New South Wales. Expanding Uruguay and the area in which slavery is abolished will be quite popular in England and may build public support that might not otherwise be forthcoming for military intervention and expansion there.
And the British do respect private property. And Uruguay is largely divided up into estancias by prior Spanish land grant. Which the British are not likely to interfere with, both because doing so would be against British law and because the local elite is the key to British rule over Uruguay remaining uncontested.
So once the Napoleonic Wars end in 1814, the British (who will now look upon Uruguay as a substitute for the New Orleans they failed to capture) will want to maximize the productivity of the land by having a cheap labour force to grow tobacco, rice and cotton, all labour intensive crops. Fortunately, India is close enough so that labour can be brought in from India by the BEIC on the return trip to England, as contract labour. The system will not be put together in a day, but given the realities of land ownership and the potential of the place, it will come together. Especially if the colony expands north to the Tropic of Capricorn and beyond.
And yes, the United Kingdom is of necessity the only real exception to the United States' Monroe Doctrine--able to take over land in the Western Hemisphere still, with impunity. Without the UK on it's side, for the US there is no enforceable Monroe Doctrine. At least not south of the Caribbean Sea.