Evasion and uncanny dodge might not be specified in the original, but they seem like a good fit (plus these have 20HD!).
Want to make the compulsion to chase it come from an aura or a gaze?
Those unlucky enough to see the creature, though, are overcome by the grace and beauty of the creature unless they make a successful save vs. spells. If the save fails, the character is so struck by the creature’s elegance that he feels he must capture it. The character must pursue the silver dog for 1d10 + 4 rounds, seeking not to injure it, but to catch it alive and unharmed. All the while, the dog flees at its full rate, heading ever deeper into the forest. At the end of the character’s monomaniacal desire to capture the dog, he may or may not continue the pursuit. Unfortunately, many such chases end with the character separated from his party and lost in deep woodlands.
Maybe instead of a gaze, we should do something like a mummy's despair ability: anyone who merely looks at it. Like this:
"Name for ability" (Su): At the mere sight of a silver dog, a character must make a DC X Will save or become obsessed with the creature's beauty and elegance. An obsessed character is compelled to chase the silver dog (which flees at full speed) for 1d10+4? rounds, intending not to harm it but to capture it uninjured. This is a mind-affecting compulsion. The save DC is Charisma-based.
I left the duration the same, but I'm not averse to changing it.
Fleeting Beauty or Lure of Beauty both sound good to me.