Elon Musk hired the high-powered Washington lawyer Dane Butswinkas to help clean up his mess with the Securities and Exchange Commission.

A day after Musk insulted the SEC in a tweet calling the agency the “Shortseller Enrichment Commission,” Butswinkas formally entered the case as Musk’s lawyer in Manhattan federal court. The tweet came after Musk agreed to settle the SEC suit, which grew out of tweets on Aug. 7 in which he talked about taking the company private. A judge...