Your distinction between "real" objects and merely mental objects in this context has certainly been pursued, as far back as Aquinas (and even further back when you consider Ockham), in his criticism of Anselm's ontological argument. From the IEP article on Anselm's Ontological Argument:
On Aquinas's view, even if we assume that everyone shares the same
concept of God as a being than which none greater can be imagined, "it
does not therefore follow that he understands what the word signifies
exists actually, but only that it exists mentally."
This could be generalized to your position as follows:
Apparently non-denoting singular terms which figure in true sentences refer to objects which exist merely mentally, not actually (or really).
Now, you aren't the first one to adopt such a view. In fact, William of Ockham held precisely this view. He postulated a "mental language" which had concepts to serve as the referents of "fishy" objects like fictional entities or abstract objects. From the SEP article on Ockham:
Ockham was the first philosopher to develop in some detail the notion
of “mental language” and to put it to work for him. Aristotle,
Boethius and several others had mentioned it before, but Ockham's
innovation was to systematically transpose to the fine-grained
analysis of human thought both the grammatical categories of his time...[and] the central semantical ideas of signification, connotation and supposition....
The "language of thought" hypothesis has been defended more recently by Jerry Fodor. See the SEP article on the language of thought. He holds, much like Ockham, that nearly all lexical concepts are innate. So, e.g., we're born with a concept of "carburetor" -- this has struck many as implausible. You're likely to have similar problems since you'll have to answer the question of where these concepts with no instances came from. See this paper on radical nativism to see some criticism from a cog sci angle.
Ok, back to Ockham. From the section on his rejection of universals:
For Ockham, the only universal entities it makes sense to talk about
are universal concepts, and derivative on them, universal terms in
spoken and written language. Metaphysically, these “universal”
concepts are singular entities like all others; they are “universal”
only in the sense of being “predicable of many.”
Thus, for Ockham, putative abstract objects like universals refer to mental concepts -- and these concepts are innate.
Quine's Solution to the Problem of Negative Existentials
From my answer to this old question about imagining nothing:
Some philosophers, notably the Meinongians, have thought that there
are some things that have the property of "not existing". So, they
would analyze negative existentials like "There are no unicorns" as
expressing the sentence "There is something such that it is a unicorn
and it doesn't exist". They could do this because they distinguished
between two senses of "there is". One, the one familiar to us from
Quine, is to read "there is" as expressing the existential quantifier.
Anything that "there is", in this sense, exists. Now, the other sense
of "there is" is subsistence. They thought that there are some things
(like unicorns, for example) that subsist but do not exist.
Quine thought that this talk of there being things that don't exist
was a bunch of nonsense. He held that "there is" only expresses the
existential quantifier and that anything there is must exist (as an
aside, he famously, but uninformatively, answers the question "What is
there?" with "Everything"). But then how did he analyze our earlier
sentence about unicorns? He would analyze is thusly: "It is not the
case that there exists something such that it is a unicorn" (this would all be regimented in first order logic for Quine).
For Quine, this sentence carries no presupposition of anything's
existence, much less of a unicorn which subsists but does not exist.
My Own Worries
Ok, now for the reason that I think a view along the lines of what you're suggesting is very problematic. How many concepts for nonexistent things are there? It seems to me that there are infinitely many nonexistent things (if there aren't already I can make up some right now: the round square with a 1 meter area, the round square with a 2 meter area,...).
Now, the problem here is that the assumption that we could hold a countable infinity of concepts in our head seems problematic -- it isn't obvious there's enough storage space ("memory"). Once you realize that there are different orders of infinity (e.g., the real numbers are uncountably infinite, making them strictly larger than the natural numbers, integers, or rationals; and things get MUCH bigger from there), the assumption that they could all be some sort of mental concept seems incredibly implausible. Even if we can hold countably many concepts, the higher up you go in the orders of infinity, the more it seems that storing that many concepts in a mind is impossible. So, you can't produce enough mental concepts to serve as referents for all of the possible terms appearing to refer to nonexistent objects.
So, maybe at this point you'd say "ok, the concepts don't have to actually be in anyone's mind it's enough that they're merely possible mental concepts". But mental concepts detached from any minds start to look a whole lot like abstract objects, and your view collapses into one of very many different possible views (all of Meinong, Frege, and Kripke could be read as holding such a view; Frege most obviously since he often used mental language to describe propositions ("thoughts"), even though he vehemently rejected this psychologistic picture).
Ok, that's all I can think of off the top of my head, but should give you some idea of where to go for further research.