The largest scale fighting in Patagonia occurred in from 1878-1883, when General Julio Argentino Roca (later President of Argentina), launched the Conquista del desierto (Conquest of the Desert).

This was a series of campaigns by the Argentine army against the indigenous people to establish Argentine control over Patagonia.

The successful campaigns are controversial, being called a genocide by some.

There were many casualties among the native Americans and many thousands were displaced from their lands.

The Chileans (who had various border disputes with Argentina in Patagonia) supplied the Native Americans with firearms, and there were several small scale clashes between Argentine and Chilean troops.

I don't know if you would call this "large scale fighting", but it was the largest scale fighting seen in Patagonia.