The
Earth - Moon doesn't orbit the
Earth - Moon - Sun barycenter exactly but it is not orbiting the barycenter of the solar sysem either; to some approximation the innermost planets and the sun must wobble around the barycenter together as they are similarly affected by the outermost planets which happen to be more massive as well as more distant and thus dominate in their effects
on the barycenter —
things should tend to get more
complicated when the planets involved are at more similar distances.