The theological claim to which you are referring is that God is changeless insofar as His character and nature are concerned. I recognize you don't believe in God and that you therefore see that theological claim as unpersuasive. Okay. But you are distorting the claim in order to create a strawman. A being that is changeless insofar as nature or character is concerned can still have thoughts or take action. To say otherwise is to redefine what Christians have been saying for 2000 years.

Nonetheless, even if you're going to unfairly and inaccurately define immutability in order to "defeat" the Kalam Cosmological Argument...

At best, that undermines the doctrine of immutability as understood by orthodox Christians.

It does not undermine the essence of the Kalam Cosmological Argument itself, which is not (in and of itself) tied to any of the monotheistic faith traditions. The KCA only establishes there is a First Cause. It doesn't tell us that that First Cause is the God of the Bible (or the Quran or any other text).

The KCA can't be "defeated" by appealing to what you see as conflicting beliefs within one of the monotheistic faith traditions.