I don't think it's as simple as you presented. If the ascribed properties of unicorns or Santas are not really properties at all, then we need a way to distinguish real properties from imaginary ones. We've gone from 'how do we know if a thing exists' to 'how do we know if a property exists'.
Now whiteness, I suppose, is a real existing property (at least of some things). New-fallen snow is white; Snow has that property; and snow is real. Santa's beard is reputed to be white, but it doesn't really have that property, since Santa doesn't exist. So the property-ness doesn't seem to help much.
For a thing to exist, does that mean it has to be perceived in some way (and not just mentally)? That perception or observation yields a (real) property of the thing, and therefore the thing exists, because it has a property?