Do you know what *immutable* means?   Sorry, but for many who define immutable, it does not mean that there is no visible change.  Geez - moral law moved from the OT to the NT; we have moved from a classical mindset to a historical consciousness mindset.  You confuse and conflate the reality that there is a core that we believe does not change but that it is expressed and lived tempered by the times, cultures, and changing concepts of human life.
Immutability does not mean that God does not vary. The incarnation is an example of variation. Also, God's attitude toward a person is changed when the person becomes a Christian. For example, the enmity between God and man is removed (Rom. 5:10).