Elizabeth referring to Mary as the mother of their Lord, i.e., their King, does not contradict the fact that their Lord was also their God any more than the Messianic title "the Son of David" does. To paraphrase John the Baptist, He who came from her is greater than her because He preceded her. Again, the Messiah is both the Root and the Branch of David, and it is being the Root which makes Him greater than all of His ancestors and puts them under the obligation to obey Him.
"Mother of God," on the other hand, suggests that Mary pre-existed the Eternal One and that He is bound by His own Law to honor and obey her--which would make her greater than God. Obviously, that's heresy.
I understand that "my Lord" may or may not, in your mind, refer to "my God", but if motherhood conferred authority to order the son around, as you claimed in 2,577, then Elisabeth acknowledged that authority, and it is in the scripture.