Does a med spa have to register or hold a license in South Dakota?

Short answer
Yes — the cosmetology side is licensed by premises: a salon or booth license costs $75 to open and $55 to renew, on top of each practitioner's own license; separately, anyone practicing medicine there must hold a board license.

South Dakota licenses the PERSON for each practice and inspects the PLACE. No person may engage in the practice of cosmetology unless licensed by the commission, and no person may engage in the practice of esthetics unless licensed by the commission — esthetics being one of the practices that constitutes cosmetology. The chapter defines a “booth” as any part within a licensed salon that is rented or leased for the purpose of rendering licensed cosmetology services as a separate, independent salon business, and an “esthetician” as any person who, for compensation, engages in the practice of esthetics but not in other practices of cosmetology. The commission, any state or local board of health, or any officer or employee of them, may enter during business hours any salon or school licensed under the chapter AND ANY OTHER PLACE where the practice of cosmetology is followed, practiced or taught, for inspection and enforcement.