If you watch Timerman's rambling presentation, it is clear that he has painted the UK as the aggressor and South America the peace-lovers. An additional 'show of force' would simply be manna from heaven in his eyes (and presumably those of his government) and provide more 'substance' to the regional argument. As I have pointed out, the present disposition of British forces is more than adequate to protect our interests. Even with our forces so thinly spread, we still have global reach and could respond very rapidly if tensions spill over to conflict. None of the other country in the region have the range of military options that the UK has (TLAM, Stormshadow, SSN, C17s, combat hardened troops &c).
Overseas territories, by virtue of their origins and management are anachronistic in many governments' eyes (and amongst many of our colleagues in the FCO) but the UK is duty-bound to protect them and not use them as a bargaining chip as Brown tried to do with the Cyprus SBAs in 2008. Luckily for all concerned, that fell flat!