Etymology of the word "fanny"

Is it not likely that the US derivation of the word “fanny” came from US Expeditionary Troops returning from France between 1917 and 1921 who saw French Pétanque players “kissing the fanny”. In Pétanque it is traditional that when a player loses 13 to 0 it is said that “il est fanny” (he’s fanny) or “il a fait fanny” (he made fanny), and that he has to kiss the bottom of a girl called Fanny. See The Art of Kissing Fanny for further details.

It would not surprise me if this was also the source of the UK meaning of fanny.

During that war, many people were literate, many letters were exchanged, and there was often a desire for something to serve as a diversion from talk of death, disease, and destruction. It would be truly astounding if your idea was correct and yet no one anywhere wrote about it.