Our account of existence/realty/ being has to account for how it both classifies things and does more than this. When I read mythology to my three-year-old and she asks me if Gorgons exist, I can’t be talked out of the idea that she is looking for a classification: she wants to know if Gorgons are things like sisters and the cars in the garage or like stories and make-believe. She wants to know if Gorgons are the things that keep existing when you’re not thinking about them (and so could get you while you were sleeping).
All the same, it’s true that existence also cannot be a classification. Classifications group like things together and so understand them abstractly, which means that all classifications are indifferent to whether a thing exists concretely. But when we call something existent we are obviously not indifferent to whether it exists concretely. One account of what Kant is driving at in denying that existence is a real predicate is that all predicates belong to multitudes and so are abstract, but existence does not belong to things as abstract.
My solution – which was the common Thomistic manual response, cf. especially Grenier and Gredt – is to see being/ existence/ the real as transcending the difference between the abstract/logical and concrete, where the transcendent is that which unifies what is diverse in lower orders. Friendships of virtue unite the pleasure and utility that are diverse in lower sorts of friendship; the brain is a sense organ that unifies sensations that are diverse in lower, diverse sensitive powers; the Being unifies both what exists dependently on the mind (the abstract, the classificatory) and independently of it (the concrete.) I don’t know how this works out in Thomistic terms (where esse goes, for example) but the more I turn over this idea I think one could teach a good part of the history of metaphysics as a failure to recognize the transcendence of being. If we have to choose between the abstract/logical and the real, we’ll end up either vindicating the logical with some form of idealism or vindicating the real with a realism that turns abstractions into ficta. If there is no transcendence of being w/r/t/ the real and the ideal, we are left trying to figure out how they can line up or interact with each other, leaving us with either a pure phenomenalism or the crudest of materialism.