Any foreign aid the UK can throw into SA is dwarfed by that which China (a country that is also best buddies with Argentina) is pouring into the country.
On top of this SA currently has a Mugabe-lite in power and could very soon have their very own full-fat version, complete with post-colonial chip on shoulder and corresponding 'international relations self-destruct button'. Any attempt to use SA as a staging post for operations in the South Atlantic would have the loons within the ANC screaming blue murder and 'imperialist aggressor' to the UN before an aircraft had left the UK.
All this is so much guff though. The Argentines are not now, nor likely to be anytime soon, in a position to use military force against the FI. Their government distrusts and underfunds the military worse than ours does. Plus the possibility of a British nuclear submarine being in the area is one hell of a deterrent to a "once bitten, twice shy" Argentine navy.
If Argentina tries to take the FI it'll be through the UN with their mates in South America, the Far East and Africa in tow.