Natural evil as necessity
The natural-evil-as-necessity argument is meant to be a response to the classic philosophical argument of the Problem of Evil, which contends that an all-powerful, all-knowing and beneficent God cannot exist as such because natural evil (mudslides which crush the legs of innocent children, for instance) occurs. Peacocke contends that the capacities necessary for consciousness and thus a relationship with God also enable their possessors to experience pain, as necessary for identifying injury and disease. Preventing the experience of pain would prevent the possibility of consciousness. Peacocke also takes an eastern argument for natural evil of that which made must be unmade for a new making to occur; there is no creation without destruction. To Peacocke, it is necessary that organisms go out of existence for others to come into it. Thus, pain, suffering and death are necessary evils in a universe which will result in beings capable of having a relationship with God. God is said to suffer with His creation because He loves creation, conforming the deity to be consistent with the Christian God.
I haven't read your post, as it doesn't interest me. But 'The natural-evil-as-necessity argument' is the argument that natural evil (whatever that is) is an unavoidable feature of the human condition. ('Necessity' has nothing to do with physical need in this context; it is logical necessity - that which must ['necessarily'] be true, given the postulates. I'm not sure from your paraphrase whether you understand this.)