Of the all fixes the SEC wants Tesla Inc. to make in the wake of Elon Musk’s now-infamous tweet, one stands out for its novelty: “An experienced securities lawyer” to review all social media communications by the company’s senior officers.

In resolving its fraud claims against Tesla and Musk, the Securities and Exchange Commission specified in the fine print of its settlement proposal that the lawyer hired or designated to vet tweets must have qualifications that “are not unacceptable to...