"...she would have been another pre-Fall Eve and therefore immortal."
But we know she wasn't merely another pre-Fall Eve; she was the Second or New Eve as Christ was the Second or New Adam.
Of course, if some Catholics are right, and she never died, then that would fit into the theory you are advancing.
Perhaps we have been looking at this entirely the wrong way. Here is what +John Cassian says in Book II Chapter II of his "Seven Books on the Incarnation of the Lord, Against Nestorius"
"Hear then what the archangel Gabriel announces to the Virgin Mary. “The Holy Ghost,” he says, “shall come upon thee, and the power of the Most High shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.” Do you see how, when he is going to point out the nativity of God, he first speaks of a work of Divinity. For “the Holy Ghost,” he says, “shall come upon thee, and the power of the Most High shall overshadow thee.” Admirably did the angel speak, and explain the majesty of the Divine work by the Divine character of his words. For the Holy Ghost sanctified the Virgin’s womb, and breathed into it by the power of His Divinity, and thus imparted and communicated Himself to human nature; and made His own what was before foreign to Him, taking it to Himself by His own power and majesty. And lest the weakness of human nature should not be able to bear the entrance of Divinity the power of the Most High strengthened the ever to be honoured Virgin, so that it supported her bodily weakness by embracing it with overshadowing protection, and human weakness was not insufficient for the consummation of the ineffable mystery of the holy conception, since it was supported by the Divine overshadowing. “Therefore,” he says, “the Holy Ghost shall come upon thee, and the power of the Most High shall overshadow thee.”"
So, while Panagia had through an abundance of grace maintained a sinful existence up to the point of the Annunciation, nevertheless she had not attained a god-like theosis, otherwise there would have been no need for the action of the Holy Spirit in strengthening her and sanctifying her womb. I would further suggest that as a human living before the completion of the purpose of the Incarnation, she could not have attained theosis borne Christ without the special intervention of the Holy Spirit.