having one classical property is enough for a thing to exist

I have it on good authority that, to put things in terms of properties, interpretations of quantum mechanics would agree that an electron possess mass and total spin (and electronic charge?) before observation. But whether it possess position, momentum, angular momentum, etc. is up to the interpretation.

I'd think having at least one classical property at a given time is enough for a thing to exist at that time.

A conceptual trick is that if a thing (such as a pair of particles in the singlet state) doesn't possess the property of position, then you can't think of it as being contained within space, before observation.

Thus, non-locality isn't a problem, and realism is safe from Bell inequalities. :-)