Changing CompositeBindings with the New Inputsystem

Hey everyone,
I am currently working on a game, where its important to be able to have different Buttons for one and the same Input Action. Especially composite. So i was trying to add a PartBinding in my Composite WASD with the InsertPartBinding and that worked. I could steer my movement with the corresponding button but it did not show up in the StarterAssets Action Asset ( I am using the FirstPerson Template). When i changed another action that is just a Button i saw the the change in the starter asset. So what do i need to do that adding another part into a composite binding shows up in the assets too? This is a problem for me cause i think its the reason that i cant find the binding later on when i try to remove the binding again see RemoveBinding function in screenshot. When i debug i can find the binding somewhere in my action in visual studio but i am not able to find the action via code or in the asset. Can anyone help me out here?

    public override void AddBinding(InputAction action)
    {
        string b = GetPathString();
        var c = action.ChangeCompositeBinding(compositeName);
        //was machen, wenn button schon anders belegt?
        c.InsertPartBinding(bindingPartName, GetPathString()).InsertPartBinding("Down", "o").InsertPartBinding("Right", "c").InsertPartBinding("Left", "p");
        RemoveBinding(action);
    }

    public override void RemoveBinding(InputAction action)
    {
        var binding = action.ChangeCompositeBinding(compositeName);
        
        for (var b = binding.NextPartBinding(bindingPartName); b.valid; b = b.NextPartBinding(bindingPartName))
        {
            if (b.binding.path == GetPathString())
            {
                b.Erase();
                return;
            }
        }
        Debug.Log("Could not find BidingPart " + bindingPartName);
    }