Yet that isn’t what was recorded....
when you limit it to referencing Exodus 3:15, alone.
For wider meanings, the greater truth of the matter, try this; http://www.hebrew4christians.com/Names_of_G-d/Adonai/adonai.html
But it is affecting the meanings --- as if Elizabeth was declaring the child to be God -- when the word she is attributed to using can also be reasonably enough interpreted (from common Hebrew usage) to hold back from that "paraphrasing".
The words used for "Lord" and for "God" do not translate uniformly across the board to make such assertion as you have.
At the Council of Ephesus, if what was aimed for would have been to refer to Mary unabashedly quote-unquote "Mother of God", then they could have/would have coined a phrase more explicitly declaring that very thing.
Instead, by adopting a term which indicated "birth-giver" or else "bearer" they bumped up against that as close as they could without going all the way.
Why the reticence on their part -- if one could now both rely upon such things, while ignoring that crucial element?
May I suggest that although it was being recognized that Mary was mother of Jesus, and that Jesus was surely enough Emmanuel (God among us), Jesus was not the Father (His own) but was instead Son of God, and in that manner and way was himself God -- in the flesh, in human form?
How limited Jesus may have been, in comparison to the One eternal God had been the basis for debate. That's understandable enough, for should one contemplate that The Father (God is a Spirit, as it is written) hungers, thirsts, requites sleep, etc? Jesus, in form of a man, did appear to require such things...yet as Jesus is attributed to having said--- nothing of the Father was kept from himself.