Australia doesn't give Britain political leverage over Indonesia, Canada doesn't give Britain political leverage over the US, South Africa is not why the UK has political leverage over the african commonwealth countries.
I think we need to talk about the difference between back in the 19th Century and post-Statute of Westminster times.
The Australian colonies were a very long way away from Indonesia. Although British control of Singapore did give them a lot of influence over wider South East Asia.
British North America did give them influence in the United States, and was a key reason cited by the Federalists as a preferable US ally over the French, despite previous history. This influence was limited by the time the US became a major industrial power however.
South Africa was a VERY big reason Britain ended up dominating much of southern Africa (private missions into Rhodesia, protectorate over Malawi etc). This would be more than the TTL South America though, because African states would have been weaker than the other South American colonies.
Argentina won't magically give more political leverage over south America than they had, which was already enormous when the empire was at its zenith, and will still fall apart to not much as the empire breaks up and various cuts are done in the 20th century.
It will give more political leverage and would likely be accepted as Britain's sphere of influence in the way it was accepted as the US's sphere of influence in OTL. However, as the US grows in power, it is likely that Northwest South America falls out of Britain's influence and into America's.
It might lead to Argentine Falklands, however.
Almost certainly.
Additionally, it's unlikely that the internal political system of british Argentina would be more than the old legal and social customs with a slight British veneer (as it is in French Canada and South Africa), probably with some legal protection of English (as with South Africa) or French (as with Canada).
It would all depend on how much British immigration there is. Remember that Ontario was part of French Canada. Even today we have a very pro-British PM in Canada from our influence there.
And the British Empire would be stuck developing the region on its own money directly. An empire can only chew on so much.
Yes, it would probably ending up costing Britain more than in OTL. However, it's hardly going to bring the British Empire to its knees, as it will be small net costs in the overal scheme of things.