When a man in a Cartier watch stepped into the IWC Schaffhausen boutique in Beverly Hills recently, salesman Hua Huynh sprang into action. He led the customer to a case of Aquatimers, the Swiss brand’s line of self-winding dive watches. “What’s the damage?” asked the customer, pointing at one model. “The value is $5,800,” replied Mr. Huynh. “Would you like to try it?”

Hovering nearby, Jean-Marie Brücker winced.

After...