Why attack it? Drilling Units are thirsty beasts consuming not only fuel and victuals but well casing, drill bits, drill collars, drilling muds, cement, perforating explosives and other exotica that only a driller could name. Supplying a single Unit remotely when none of the neighbouring ports will entertain your supply vessels is bad enough and means a long supply chain to the Falklands. If oil is actually found in commercial quantities and fixed platform(s) or floating production vessel(s) are needed on location then your logistics headache - and thus your costs - can only increase.
If you do start commercial production, how do you get it to shore? Pipelaying in those waters would not be the easiest job and perhaps no South American country within range would permit landfall of the pipe and construction of the onshore storage/processing facility. That means landfall would have to be on the Falklands themselves requiring tankers to ship it out to the nearest friendly country - USA? South Africa? If a pipeline is not feasible then a FPSO will be needed (more costs - more logistics problems) and a shuttle tanker fleet for a very weather dependent operation.
Oil companies are not fools; they will have figured out all of the above and done their sums and, so far, it must look a reasonable bet or operations would not have started. However, the Earth's crust is peppered with wells that have been expensively drilled and then plugged and abandoned because either the hole is 'dry' or the quantities found are uneconomic.
My point is that Argentina and their supporters would not necessarily have to go anywhere near the drilling operations to stifle it, they could simply deny port access to the logistics back-up and wait to see if the economic arguments finish it off.
My apologies to all the Gung-ho types who want to smell cordite but you may be disappointed.
Barry M