They were being invaded, but it's a very small territory population wise, and is located far from the home islands. Would it be fair to say that in a way it was a last gasp of maintaining the British Empire?
I do not believe it was so much about a last grasp attempt to save the empire. I think there's two simple explanations.
Firstly, a foreign power invaded British territory. To leave that unanswered would have been an extremely strange decision to make, especially when considering the invading force was inferior to the British forces. If Britain had let it go unanswered without military response, especially with the back drop of the cold war, they would have appeared extremely weak.
Secondly, Thatcher was fairly unpopular by the time of the invasion. If she didn't react it would have certainly made her even more unpopular, especially amongst Conservative Party members. Whereas a swift and decisive response would help improve public opinion of her, which is exactly what happened. Swift and decisive wars very often improves a leaders popularity. Again, in the midst of the cold war, a strong response from Britain would make Thatcher appear a stronger leader in the eyes of other world leaders (especially important being a woman in what was at the time a very male dominated political landscape).
When we review events of the post war era, it is always worthwhile looking at it through a cold war lense. The cold war and a fear of communism was very influential in foreign policy.