There is some linguistic overlap with existence and being. In short, existence is not everywhere reserved for what is finite. Your distinction may still hold, but even in common parlance, being can be said to exist. And in Aristotle, the discussion proceeds on the various meanings of being, including the being of the finite and contingent. To proceed any further, it is incumbent to first separate the linguistic problem from the metaphysical.
That seems to the the case, at bottom. Do you have a proposal that can help us see how to accomplish such separation?
Unfortunately language is necessary to convey concepts to other minds. But unfortunately language performs its own "reduction" of reality in order to communicate insights about that reality. This is a seemingly insoluable paradox.