French settlement of the Falkland Islands was brief and limited, and lacks continuity with British settlement. It seems unlikely that the English name should be derived from the French Dauphin.
If the name isn't derived from a habit of feeding on dolphin carcasses, an alternative explanation is that it might be named after HMS Dolphin:
"Between June 1764 and May 1766
Dolphin completed the circumnavigation of the globe. This was the first such circumnavigation of less than two years.
[3] During this voyage, in 1765, Byron took possession of the
Falkland Islands on behalf of Britain on the grounds of prior discovery, and in so doing was
nearly the cause of a war between
Great Britain and
Spain, both countries having armed fleets ready to contest the sovereignty of the barren islands."