I really like both Alice and Daisy (in fact, have used both!) but I'm not entirely convinced they go together. I think one name needs to be three syllables to balance the shortness of Alice or Daisy, personally. Alice Rose too short overall for my ear.
I think Alice Daisy works. Any anyway, how often do people get referred to by their first and middle name? If the middle name is getting thrown in to conversation most likely the surname will too, and that changes the whole dynamics again.