Pretty sure cake meant bread in the original quote.
For one thing, the original French phrase that Marie-Antoinette is supposed to have said—“Qu'ils mangent de la brioche”—doesn't exactly translate as “Let them eat cake.” It translates as, well, “Let them eat brioche.”
Brioche is a light and fluffy naturally sweet bread, often had with coffee.
On that note, Bourgeois means "middle class" in a literal translation to English, but it refers to "middle class" in an English sense rather than an American one. Doctors/Lawyers/Business Managers instead of a catch-all that anyone can self-identify as.