The problem with your argument is that you assume that changeability is the heart of the dilemma. It is not.
The real issue is that the proposer of a God-based morality wants to be able to meaningfully say that God is good in a non-trivial sense. But they also want to deny that there is any standard by which we could say such a thing. For example if we tried to ground morality in terms of God's nature we would have to say that if God's nature were such that he approved of child rape, child rape would be good. And this applies to any and all moral commands. .
To refute this it must be argued that God's nature must be such that God would necessarily approve of "good" actions and disapprove of "evil" actions - without setting up a standard by which to judge an action "good" or "evil". But how could this be done ?
Ultimately, most people who propose God as source of morality also tacitly assume that there is an independant standard of morality. They appeal to God's nature on the assumption that God's nature is necessarily good in an objective non-trivial non-question-begging way. Without this assumption all we have is an arbitrary redefinition of morality which can - and should be - rejected.