I think the reason you do not have those options appearing on the Pet Palette when Hidden Morph Geometry is displayed is because in that case you can edit that Hidden Morph Geometry. So you can edit the sub-polygons of Morph Faces but not the full Morph Faces themselves. In this case all those commands of the Pet Palette that apply to full Morph Faces (Push/Pull, Tube, Bulge, Offset All Edges, Custom Face Settings) are not displayed since those do not apply to Face sub-polygon editing.
This is a feature of the Pet Palette that it displays only those commands that make sense in the given situation. For example, when you have your cursor on a polygon node (vertex), the "Offset Edge" command is not available in the Pet Palette because it does not apply in that case.
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