As far as I can remember, I have almost always been asked to show a form of identification when I have paid with a credit card. The only ID I carry with me is my FL driver's license since I also leave my passport in a secure place when visiting Peru. I have not encountered a situation where I couldn't pay without showing my passport - unless it is something new (last time I was there was in November).
I wonder if a photocopy of the passport, along with the DL would be enough to convince them that the card its actually yours. Maybe I'll try that if I go to Falabella when I go back within the next month or so.