It is difficult to analyze existence as a predicate, property or reference. It may or may not be a property, it might be a property of realities whose inverse is only a property of fictions, etc... And an overall theory of opposites requires doing so. So coming at the question from that angle is hard work probably not worth doing.
Instead, before you get to existence, just realize that one or the other properties your object definitely has probably already has the same diseases that 'existing' does, and those flaws are easier to see.
There are many dimensions along which to contrast two things, and many of those dimensions do not have a single sense of opposition.
Location in space, for instance: Your opposite should not be here. Could it have its center an infinitessimal distance away? Would that be 'not here'? Do the opposite positions in space at least have to not overlap? Maybe 'here' is two inches to my left. Then is the opposite two inches to my right? Is it the position on the opposite side of the Earth from the point two inches to my left? Is it he opposite position relative to some centerpoint in space?
Color: Maybe it is pink. According to what color wheel or radiation composition chart should I choose the opposite color? Is it forest green, or does it predominantly emit some wavelength distribution that is made up of the reciprocal energies to those that make up red light diffused with white?
Time: This thing exists now. When is the opposite of now? Can this thing's lifespan and its opposite overlap and still be opposites?
Only concepts have opposites, a single actual referent involves too many conceptual frameworks to avoid having some of those not defining opposition.