Errors calculating clip-space Z position on the CPU side

I'm trying to write a custom outline shader. I went off of an example that gave me a perfectly fine outline, but the outline is designed to always be in front of everything except the mesh it's outlining. I wanted to change this to make the outline basically sit just behind the object itself so that it can be occluded properly. I did this by adding the following property to my shader:

 _ClipSpaceZ("Clip Space Z", Float) = 10.0
 
 //...later, in CGPROGRAM block:
 
 float _ClipSpaceZ;
 
 //in the vertex shader:
 
 o.vertex = mul(UNITY_SHADER_MVP, v.vertex);
 o.vertex.z = _ClipSpaceZ;// * o.vertex.w

I set the value of this property in a custom script. The user defines a bounding sphere that encloses the mesh being outlined, and every frame a position on that sphere is chosen opposite the camera's position (i.e. behind the mesh from the camera's point of view). The clip-space Z coordinate of that position is then passed to the shader.

 private void Update()
  {
      Vector3 boundingSpherePos = outlineMeshTr.position;
      Vector3 camToSphere = (boundingSpherePos - camTr.position).normalized;
      Vector3 worldOutlineCenter = boundingSpherePos + (camToSphere * BoundingSphereRadius);
      Vector3 clipOutlineCenter = (CamObj.projectionMatrix * CamObj.worldToCameraMatrix).MultiplyPoint(worldOutlineCenter);
      outlineMeshRenderer.material.SetFloat("_ClipSpaceZ", clipOutlineCenter.z);
  }

However, this doesn't work properly; the outline is basically always in front of the mesh. How can I correctly calculate and use the clip-space Z position of worldOutlineCenter?