If “God” ALWAYS means “the Trinity,” then the statement
“The Father is God.”
means “the Father is the Trinity.”
Is that what you mean when you say “the Father is God”?
“The Father is God.” is not problematic because it is a narrowing reference to one of the three persons, and is structured to focus on what all three persons have in common, which is the divine essence. And that's fine, if that's what you mean by it, but to avoid the problem of the undistributed middle, you must use the same term the same way in both the major and the minor premise. This applies to using "mother" in a generative sense as well. If we reformulate your syllogism using this principle, we get this:
Premise 1: Jesus is God in terms of the divine essence of His being.
Premise 2: Mary is the Mother of Jesus in the generative sense
Conclusion: Therefore Mary is the Mother of God in terms of generating the divine essence of His being
All we are trying to do is be consistent in how we use the terms. Obviously, the above syllogism is false. You can try to rescue it by using the term "God" inconsistently, but then you no longer have a valid syllogism. Sorry.
Peace,
SR