My work around is to put the WP generated body classes from body_class() on a div inside the #site-wrap div I am using in the "Content Element ID" setting.
Note that anything (such as classes) on the element you're using for the "Content Element ID", and outside of it doesn't change.
So if possible for some of you, it is easier to move body_class() from the body element inside what is actually being "refreshed" on the page so that it gets updated and can be used for CSS and PHP, rather than trying to change anything on the body tag.
(this is only for documentational purposes)
Original question
Solution - please see answer of "Courtney"