If God cannot change, then why does His wrath need to be appeased? More or less I am asking for a defense of the propitiatory view of atonement in which God changes from being angry to being happy with us rather than sinners being the ones who need to change. Perhaps I misunderstand this model.
You misunderstand the model.  But it's a common misunderstanding.  One that comes up often enough that there's a standard answer from Apologetics.  Here is is in my words:

The statement that God doesn't change means that His nature doesn't change.  His Nature includes many attributes that never change:  Goodness, holiness, He is Just, Righteous, and executes Judgement.  None of these change.

His attitude toward us can and does change.  This is not only consistent with the idea that His nature never changes, it is necessary if His nature of righteousness never changes. If God is righteous, He must punish sin.  Therefore, if we are in sin, He must punish us.  If He is forgiving, then His attitude toward us must be able to change from wrath to forgiveness.

I am a parent.  I love my children.  I will always try to do what's best for them, which means I will praise them when they deserve praise, I will punish them when they deserve punishment, I will comfort them when they are hurting, and I will burn with anger if anyone hurts them. 

The fact that I can be angry with them when they are naughty and please with them when they are good does not mean that I change.  I remain the same.  Who I am does not change, my nature does not change. Neither does the fact that God can be angry with us, and then forgive us means that He changes.

More on this at Apologetics Press, on the question of "Does God repent?"

Excerpt:


  For instance, during the Patriarchal Age in which they were living,
  Noah and his contemporaries had received instructions on how to live
  righteously (see 1 Peter 3:18-20), and as long as they continued in
  this manner, God’s presence and blessings would abide with them. But
  when they became sinful and unrepentant, He no longer could condone
  their actions. As a consequence of their sinful rebelliousness, God
  withdrew His spirit (Genesis 6:3), and pledged to send a flood to
  destroy all mankind except Noah and his immediate family (6:7). God
  was grieved (6:6), not because He did not know that this series of
  events would happen, or because He somehow “regretted” having created
  man in the first place, but because, having given man the choice to
  serve Him or reject Him, man had chosen the latter with such
  unanimity. When we hear God described in terms such as “sad,”
  “joyful,” etc. that frequently are used to describe human emotions, we
  must remember that such descriptions are not intended to imply that
  God is emotionally vulnerable in the same way that humans are (cf.
  Acts 17:25). Rather, such descriptions are intended to show that God
  is compassionate and loving.