PN, both!
i don't believe there is a 'good' course of action available, there are merely a number of crap options with differing downsides.
i do not believe that any real headway can be made into the Argentine political psyche, i think we just have to live with the fact that they will always want the Islands, and that any Argentine politician who ramps up the rhetoric will be rewarded electorally.
given that, we have three options - to maintain a minimum defence that doesn't offend the other LA nations, and to attempt to use economics and diplomacy to ensure that LA solidarity on this issue stays as a very thin concept.
to ramp up the defence and running costs of the islands to the extent where all of LA could be based at Rio Gallagos and they'd still never get within 100 miles.
to give up the Islanders to independance and the rather shaky 'protection' of the UN.
Option 1 is preferable, but it carries risks - the diplomacy may not work, and the 'token' defence may have to be so 'token' in order to be friends with the other LA countries that the Argentines are able to get through it.
Option 2 works, but is hellishly expensive, and it probably means cutting off ourselves - and the EU - from LA and most importantly Brazil, while that situation remains.
Option 3 is pretty rubbish, it solves our problem, but i don't think the Islanders will feel that safe...
personally, i'd go for somewhere between 1 and 2, with 2 being the default option. do i believe that Brazil will go to economic war with the EU to make Argentina feel good? no, i don't, but i also don't underestimate how politically dominant Brazil would be within LA if it went hardline on this issue in the name of 'regional solidarity'.