You Ask: Name of Louis XIV Papillon. (History of Papillons)

Your question of what the name is of King LouisXV Papillon, needs a little bit of further historical explanation:
It is thought that both Marie Antoinette and Madame de Pompadour helped the Papillon become fashionable and they both admired and owned this breed.

Madame de Pompadour

 Madame de Pompadour was the mistress of King Louis XV, and owned Inez and Mimi which were her two pet Papillons.
Marie Antoinette had a Sevres porcelain model of a Phalene (Papillon with drop ears) in her room that was only found after her death..

Marie Antoinette

Marie Antoinette is said to have walked to the guillotine clutching her small dog under her arm  . 
Marie Antoinette's dog was a small spaniel that had been brought to the French court from Spain on the back of pack mules. According to the story, her pup was spared and cared for in a building in Paris still called the Papillon House.
Marie Antoinette's dog was said to have descended from a very old drop-eared breed known as the Epagneul Nain Continental, or Continental Dwarf/Toy Spaniel that appeared in church frescos and paintings as early as the 13th century .
 Her house in Paris, France is known today as The House of the Papillon.

 Toward the end of the 19th century, breed fanciers bred a version of the spaniel whose ears stood up. This dog was said to have been nicknamed papillon based on the impressively large, erect ears that resembled the wings of a butterfly. The drop-eared variety of the breed came to be called the Phalène (which means "night moth"). Both types may today appear in the same litter..