Why Bismarck's last action has no single battle name

Is there an official name for the battle in which Bismarck was defeated?

Not really; it is usually described within the hunt for Bismarck or simply as the sinking of Bismarck.

The final engagement does not have one widely accepted standalone name. Accounts usually place it within the hunt for Bismarck or label the event by the ship's sinking. The operational campaign and the terminal battle are therefore often discussed under the same heading. Those search terms provide the clearest route into the event's historiography.