Fished a bit in the early 80s. The mullet fishing in Port Stanley was fantastic. I used a big float, big hook and a lump of bloody mutton, the locals called "365" At that time cattle were too valuable to kill for beef, they only ate mutton and they had it every day. I expect someone has latched on to the fact that mutton is a good bait and it's now called "Ovis aries HNV" and it costs a fortunte. We anglers got it free from the cookhouse. We enjoyed fresh fish and chips. My biggest Mullet weighed in at 5lbs but there are bigger ones there. All bays had fish in them.
The Sea trout fishing was also fantastic, we had a fishing mad boss, a flight of helicopters and plenty of time on our hands. We would be dropped at a nameless river with a 24 hour ration pack and spinning gear. We only had spinning gear, fly fishing might be tricky as it gets quite breezy down there. We had a great time and the officers mess had Sea Trout for supper. I was happy with my fish and chips.
I know how much it costs to fish there now and I expect the random fishing that we enjoyed is no longer available. Are you flying Crab Air?