I believe that your argument is unwittingly placing the eternal God in a temporal frame. An unchanging God in time would be a God trapped in amber, unable to so much as part a Red Sea. A changing God would be a God that (in changing) was heading toward or away from perfection. Either way, He would be imperfect, and therefore not God.
The Catholic God is not trapped in an instant of time **or **constantly changing. He is in full possession of himself. As a human, I am not in full possession of myself. Yesterday is inaccessible, tomorrow unimaginable. I gain or lose aspects of myself as time passes. On the other hand, God is in full possession of himself, such that parting the Red Sea, sending His Son in the Incarnation, and the Second Coming are all part of His eternal Now.
God is able to act in time without being changed because He himself is not in time, but its Creator and Master.
Frank Sheed’s “Theology for Beginners” is excellent. Pardon this extended quote, Sheed expresses the thought with great clarity:
"God is changeless because he is infinite. He has all perfections. He cannot lose any of them, so there is no past into which they can flow away. Nor is there any future from which new perfections can flow to him. He has all perfections, in the present, a present which does not change and does not cease. That is eternity. The universe he created is not like that. Things come and go. Change is continuous. Time and the universe started together.
We must concentrate upon the concept of eternity; it brings us deep into the meaning of God. You and I and all of us are in time, which means that we are never at any moment the whole of ourselves. What we were last year, what we shall be next year, all belong to our total being; but last year has gone and next year has not arrived. There is never a moment when we are all there. We possess our being, the philosopher say, successively. Not so God. All that he is, he possesses in one single act of being. Eternity does not mean everlasting time, time open at both ends, so that however far you go back into the past there is no beginning, however far you go forward in the future, there is no end. Eternity is not time at all. It is God’s total possession of himself."