What does South Dakota prohibit in med spa advertising?

Short answer
Only truthful claims — advertising of medical business that makes untruthful or improbable statements, or is calculated to mislead or deceive the public, is unprofessional conduct and grounds for the board to limit, suspend, or revoke the physician's license.

South Dakota's advertising ground is one of the broadest in this reference and it does not require falsity: unprofessional or dishonorable conduct includes all advertising of medical business in which untruthful or improbable statements are made, or which are calculated to mislead or deceive the public — “improbable” and “calculated to mislead” reaching claims that are not provably untrue. Two neighbouring items cover the sales practices around it: obtaining any fee on the assurance that a manifestly incurable disease can be permanently cured, and employing what is known as cappers or steerers.