Not quite what you were looking for, but it was apparently a "thing" back then.

This is from Pliny the Elder's Natural History:

“It is generally believed that ass milk effaces wrinkles in the face, renders the skin more delicate, and preserves its whiteness : and it is a well-known fact, that some women are in the habit of washing their face with it seven times daily, strictly observing that number.

Poppaea, the wife of the Emperor Nero, was the first to practice this; indeed, she had sitting-baths, prepared solely with ass milk, for which purpose whole troops of she- asses used to attend her on her journeys.”

I couldn't find specific quotes regarding Cleopatra though. The fact that Pliny said Poppaea was the first to practice this would imply he was unaware of Cleopatra doing it, since she had died ~70 years before Poppaea was born.