As a simple mutable solution you can call splice twice in a row:
playlist.splice(playlist.length - 1, 1, ...playlist.splice(INDEX_TO_MOVE, 1))
On the other hand, a simple inmutable solution could use slice since this method returns a copy of a section from the original array without changing it:
const copy = [...playlist.slice(0, INDEX_TO_MOVE - 1), ...playlist.slice(INDEX_TO_MOVE), ...playlist.slice(INDEX_TO_MOVE - 1, INDEX_TO_MOVE)]