A death row inmate who participated in the Special Olympics as a child wanted to be executed. But a lawyer—not his lawyer, another one—wanted to save him.

The U.S. Supreme Court Oct. 29 effectively took the inmate’s side in the unusual dispute, permitting Rodney Berget’s execution to go forward. South Dakota officials killed him that night.

Berget had testified at sentencing that he knew he was “guilty of taking Ronald Johnson’s life,” referring to the victim corrections officer he killed...