Is the first draw-over your desired result? I'll assume it is.
It looks like you want the outside surface of the object to remain mostly unchanged while the inside surface expands and grows inwards? That is going to significantly increase the volume of the tissue. You could try setting "volume conservation" and "Poisson's ratio" to zero on the zMaterial nodes and "compression resistance" to zero on the zTissue node. These attributes all try to make the object conserve volume, which is going to be fighting against what you've drawn.
I think you're trying to get material properties something like in the image below. This started as a nice vertical box. The left side if very stiff, the right side if very soft. Once the sim starts, it tips over and sags under gravity, but the stiff side stays stiff. You can download this scene from here. It requires a lot of resolution in the tet mesh to get this sort of material gradient through a thin object. In this scene, I've used surface refinement on the tet mesh to get extra small tets near the surface reduce that cost a bit. You may be able to do it with less, but I haven't experimented further.
If you want a really big gradient in stiffness material between the inside and outside of a thin object, you may get results more easily by using two bodies instead of one body with a spatially-varying material. You could consider using a stiff cloth shell on the outside for the "stiff parts", with a soft stretchy tissue on the inside. The other way around might work just as well or better - with a stiff tissue shell and a stretchy cloth attached inside. These will probably run faster and be easier to tune than the super high-resolution tet mesh needed to simulate a large stiffness gradient.
If these simulations are supposed to be like fat and muscles, then there's another thing we do here that might help you too. That is modelling the inner-surface of the fat so that is starts right next to the muscles. If everything starts pretty-much in contact at the start of the sim, then nothing has to grow and expand to fill the voids between bodies. This allows us to leave all of the material properties at something biomechanically realistic - more like the default zMaterial attributes. In this case, a large stiffness gradient isn't required to keep the shape and a lower resolution tet mesh works fine. In real life, skin is stiffer than the underlying fatty tissues and sometimes that matters, but we tend to model that with a stiff cloth shell rather than a material gradient.