picture of a Barbie doll

Definately something to stay away from doing with respect to IP rights. Even if in the public domain, it would not be permitted. From what I gathered by way of an internet search, Mattel will go to great lengths to protect the brand. From the internet:

"Mattel equally protects Barbie’s distinct brand identity, including its name, logo, and slogan, with its trademark. Initially registered in 1959 with a simple pink sans-serif font, the logo underwent five changes before reverting and continuing with the original in 2009. Originally, Mattel secured trademark protection for “Barbie” specifically for dolls, which subsequently extended to other product categories. It currently enjoys a wide range of Barbie-branded merchandise through licensing agreements, including Gap’s Barbie apparel, OPI Barbie nail polish, NYX cosmetics, Xbox controllers, Burger King sauce, Crocs, UNO cards, and more, all infused with the iconic ‘Barbie pink’ (also trademarked!)."