Rather late to worry about 'the political side of the conflict' when you're already in a shooting war - and it's an easy sell, too - a necessary action to protect 'our boys' to the best of our military ability. If a bunch of 'tin pot' little nations (which is how they are / will be perceived) get a little huffy, that's tough titties. Seriously, in a world where this is actually possible, it happens.
There's always time to worry about it, and it should always be thought of. Don't forget multiple South American nations were, if not activelly, at least passively and politicaly on the side of Argentina. There were background talks in the UN.
If the UK starts to up the body count on operations that the political /social world might very well see as unnecessary ("we thought you wanted just to free the islands. Now you bomb Argentina?!"), not only the preassure on the UK will increase, the other SA nations might pitch in to help "defend against more european colonialism"). And, if as some here have writen, the UK attacks Buenos Aires then there
will be serious trouble, casualties or not. Regardless of the actual results, Argentina can come up with stories of "dead civilians bombed (children!!)", civilians attacked, we shot down a bomber, etc...
The UK owns the space and water above and around the Falklands, which is their objective. Sticking to their mandate will allow them to difuse further international problems, regardless of short term military issues.