Surely it won't have escaped the notice of Argentina that if oil is found in quantities worth extracting, her best option is to seek lucrative support contracts and licenses for oil refining on the mainland. Perhaps a fixed percentage of every barrel sold?
Las Malvinas / the F**kland Islands are such a barren wasteland that most things of any worth have to be flown in or sent by sea. So there is zero chance of the place being able to support major oil industry requirements - and don't forget that both the islanders and woolly-hatted busy-body envirofundamentalists would be up in arms if the seals / penguins or other wildlife were to be forced to move out so that oil installations could be built on the islands.
A solution which benny-fits everyone is what's actually needed - jumping up and down about sovereignty claims or the possibility of Argentine military action is not going to help anyone.