Mary was the mother of Christ ONLY in His humanity not, as you stated, in His eternal deity.
To say Mary is the mother of God is completely false. Christ existed prior to Mary. It is completely wrong to insinuate that Christ was something less than He was prior to His human birth.
Nobody ever said that Mary is the origin of the eternal, infinite, Triune God. Thus, you are railing against a belief that no one has ever held or taught.
Is the following syllogism formally valid or invalid?
Emily is the mother of Sam.
Sam is a fireman.
Therefore,
Emily is the mother of a fireman.
So you are Solomon-like and want to split the child in two?
The Word became Flesh, as Jesus was both God and Man at the same time, not two different beings.
If Jesus was God and Man, Mary was His only Mother. This does not make her divine, but it does make her the Mother of God, just as Jesus was known as the Son of Man.
We should adore, honor and respect Jesus for being so humble and doing the will of the Father, and we honor and respect Mary and Joseph for following the will of God and raising Jesus so that we could save us from our sins.