I don't see why proposing a referendum by the Falklanders would be a bad idea; simply offer to make the result binding in all parties. We promise to hand over the if 50.1% of the population want it, the Argentineans shup up if 50.1% (or more likely, 99.9%) of the Islanders want to stay under British sovereignty. What could be fairer than that?
Argentina is a strange paradox, though. I was in BA last year and found them all very welcoming and, indeed anglophile. The backdrop of the "Malvinas" is everywhere (even their tv weather forecasts include the temperature for Stanley) but it was never brought up in conversation by anybody I met.