There is one sentence that says the firmament is the sky. And several sentences that refer to the Firmament Of The Sky. Using your analysis, the sentences should be interpreted as the “Sky of the Sky”.
I can’t believe I’m going to have to give this example… but it has come to this:
If someone were describing the various geological ages of the Earth … starting with the Cretaceous… and he or she said the Cretaceous period is Mesozoic… would you be justified in saying Mesozoic = Cretaceous?
This is just so obvious… and yet you have succeeded in making a mountain out of a molehill. Cretaceous is Mesozoic because it is part of the Mesozoic group of ages. Otherwise, a historian would not be able to coherently speak of the “Cretaceous of the Mesozoic”.
The scribe of Genesis 1 describes the firmament as the sky. But then he goes on to describe the Firmament of the Sky multiple times… making it pretty clear that the Firmament is a part of the sky … it does not comprise the entirety of the sky.
This should be intuitively quite clear to you, even as you pursued the idea that the Firmament = the sky = the clouds. Are the Clouds the sky? They are part of the sky. The clouds of the sky is a coherent phrase. It is clear that neither the Firmament, nor Clouds, make up the entirety of the sky.