Britain still got trade rights and economic control over the region. They didn't get their colony, but you want that more than they did.
the only way to control BA from M is to back a war lord/faction, and give him enough support to stay in power. What that gives you is trade rights, which they can get anyway without all the resource drain - either by returning the cities back to Spain in exchange for trade, or by setting up an independent country. But that doesn't give you your colony.
I still say Britain will quickly realize it's not worth the resource expense. As in OTL, they can get what they want without physical possession. Perhaps they can affect a smoother transition to independence, so that Argentina doesn't go through the massive civil disruption/wars that it did OTL. that would be a victory in itself. But that doesn't get you your spot of British pink on the map.
To have a colony be appetizing, it needs to be profitable, which means it needs stability. the colonials were getting ideas of independence prior to 1806 - it hadn't erupted yet, but the rumblings are there - and that was with a Spanish king. they sure as heck are going to be restless with a British overlord, whether it's directly or as a puppet state. Your biggest hurdle is motive. They can achieve trade without a protectorate and/or an Uruguayan colony and all the expenses involved. that's my main disagreement with your plan. Hand Wavium provides the motive, and the ability is there. I'd stick with occupation of both BA and M. Alternately, I'd go with an attempt at installing a king on a throne and forget about occupation. Go with creating a strong monarchy who will keep peace and foster economic growth. Create an ally rather than a protectorate under your thumb. It's a radical notion for the time (actually helping someone for future gain over abusing them for immediate gain), but history has shown that strong economies foster strong trade.
I think the following could constitute an attempt to create common ground (and I've based all my premises mentioned in this thread on that): Even though the new British Foreign Secretary, Viscount Castlereagh, made a famous memorandum in April-May 1807 that Latin America should be subject to British economic but not political influence, this would hold everywhere in Latin America only if the British subsequently lose in the River Plate (and, by extension, Patagonia), which is what transpired in OTL. My thinking is that if the British do make a victory in BA in June-July that year under Whitelocke, the Castlereagh memorandum would have still been applicable elsewhere in Latin America but not quite as much in the River Plate.
What this means is that inside the River Plate, under those circumstances, trade would have been
a major goal for the British but not necessarily the only one. Elsewhere in Latin America, of course, trade would have remained
the major goal for the British. Thus, in my opinion, it
would have been worth the administrative and other expenses for the British in maintaining their Plate colony.
At first, the British would have had
all the Plate, but most probably they would have gotten tired of BA and would have granted BA independence under British suzerainty, probably supplying BA with a new king from the Iberian or French lines. British forces in BA would have been triaged to Montevideo and join the British forces already in Montevideo. BA would have then been supervised from the naval bases in M and elsewhere along the Uruguayan coast across from BA, which would automatically have cut down on administrative/military costs in having to maintain bases on the BA side. If
afterwards British forces also tire of Montevideo, which I don't see as so likely, then the British concentrate their colonial activities in Patagonia, at first starting with relatively little but picking up steam in subsequent decades.