How can I rename my creature's model's hierarchy without breaking animations?

In general you still can't unfortunately. (At least not that simple see below).

The AnimationClips are based on strings storing the relative path from the Animator to the according GameObject the type of the according component and finally the name of the animated serialized fields and properties.

If any of those change e.g. because you renamed the object or change the hierarchy in general the connection is lost and the animation breaks.

You could implement an editor script method that

• goes through the affected Animator (GetComponentInParent) of the object
• iterates through all used AnimationClips
• iterates through each clips property bindings
• redirects the property path accordingly to your renaming

This could look somewhat like this

private static void RenameObject(GameObject gameObject, Animator animator, string newName)
{
    if (!gameObject)
    {
        throw new ArgumentException("No object provided", nameof(gameObject));
    }

    if (string.IsNullOrWhiteSpace(newName))
    {
        throw new ArgumentException("Object name may not be empty!", nameof(newName));
    }

    if (!animator)
    {
        throw new ArgumentException($"Selected object {gameObject} is not a child of an {nameof(Animator)}!", nameof(gameObject));
    }

    if (gameObject.transform == animator.transform)
    {
        return;
    }

    // get the relative path from the animator root to this object's parent
    var path = AnimationUtility.CalculateTransformPath(gameObject.transform.parent, animator.transform);

    if (gameObject.transform.parent != animator.transform)
    {
        path += "/";
    }

    // then append the old and new names
    var oldPath = path + gameObject.name;
    var newPath = path + newName;

    // get the runtime Animation controller
    var controller = animator.runtimeAnimatorController;

    // get all clips used by this controller
    var clips = controller.animationClips;

    var changeableObjects = new List<Object>(clips.Length + 1) { gameObject };
    changeableObjects.AddRange(clips);

    Undo.RecordObjects(changeableObjects.ToArray(), "Change animated object name");

    // Go through all clips
    foreach (var clip in clips)
    {
        var floatBindingInfo = new List<AnimationFloatBindingInfo>();

        // Get and store all FLOAT keyframe bindings
        foreach (var binding in AnimationUtility.GetCurveBindings(clip))
        {
            var curve = AnimationUtility.GetEditorCurve(clip, binding);
            var curveInfo = new AnimationFloatBindingInfo(binding, curve);

            ReplaceBindingPath(curveInfo, oldPath, newPath);

            floatBindingInfo.Add(curveInfo);
        }

        var objectBindingInfos = new List<AnimationObjectBindingInfo>();

        // also do the same for all reference keyframe bindings 
        foreach (var binding in AnimationUtility.GetObjectReferenceCurveBindings(clip))
        {
            var curve = AnimationUtility.GetObjectReferenceCurve(clip, binding);
            var curveInfo = new AnimationObjectBindingInfo(binding, curve);

            ReplaceBindingPath(curveInfo, oldPath, newPath);

            objectBindingInfos.Add(curveInfo);
        }

        // a little check to avoid unnecessary work -> are there any affected property curves at all?
        if (floatBindingInfo.Count + objectBindingInfos.Count > 0)
        {
            // Now erase all curves 
            clip.ClearCurves();

            // and assign back the stored ones
            AnimationUtility.SetEditorCurves(clip, floatBindingInfo.Select(info => info.Binding).ToArray(), floatBindingInfo.Select(info => info.Curve).ToArray());
            AnimationUtility.SetObjectReferenceCurves(clip, objectBindingInfos.Select(info => info.Binding).ToArray(), objectBindingInfos.Select(info => info.Curve).ToArray());

            EditorUtility.SetDirty(clip);
        }
    }

    // finally rename the object
    gameObject.name = newName;

    EditorUtility.SetDirty(gameObject);
}

Since this use case is quite common I took some time to implement an EditorWindow for this. It is still a bit raw but works and supports also undo redo ;) You can find it here

-> Select the object in the Hierarchy -> right click -> "Rename safe for Animator"

You could of course add some shortcut to it etc. that's up to you ;)

Here a little demo of the dialog in action, renaming some nested objects and also performing some undo/redo

However, an alternative in your use case to simply get your code to work with the names as they are might be using tags instead.

As I see it your code is based on three different cases so you could simply have a tag for each like e.g. Head, Arms, Legs and assign and check those accordingly (GameObject.CompareTag) and not touch the names and animations at all.

I have 2 plans.

1. Create an empty GameObject under the node you want to rename, and attach the collider compoent on it.

   CATRigHub001Bone004
    └ CATRigHub001Bone004Bone001
       └ Rig  <-------- Collider
   

2. Rename the bone in editor and create a script to automatically rename it to its original name while playing.

   public class Rename : MonoBehaviour
   {
       public string boneName;
       [NonSerialized] public string partName;
   
       void Awake()
       {
           partName = name;
           name = boneName;
       }
   }