“The Shulchan Aruch rules that one may not heat up water on Yom Tov in order to wash his entire body, since doing so is not shoveh lechol nefesh.”
So if washing ones entire body IS shoveh l’chol nefesh the entire issur vanishes. And as Rabbi Broyde wrote, “Halacha recognizes that the concept of shaveh lechol nefesh is determined by a societal and social norm and not by a fixed and immobile rabbinic decree” and he brings sources that prove this.