Hi Mike,
There is currently no straightforward way to do this.
Changing how the Cloth Thickness works would be a very large work package.
Unless I a missing something, I do not think your Morph idea will work because Inner Fatness works in reverse to Outer Fatness, i.e. when you have a thin geometry like a grid and use that in a Morph or Collider, the Inner Fatness will basically suck everything in that comes in contact with it.
The best approach might be to actually have two cloths and bind/stitch them to each other.
So a thin one with thickness of 0.5 * distance your red line has from the black line.
And a thick one with thickness of 0.5 * distance your green line has from the black line.
Your cloths would meet at the black line,
So do a stiff bindings/stitching and use a Filter to filter collision between the Cloths.
I tried to roughly draw in dark red and dark green where the cloth meshes would actually be.
Not sure if this is something that would work for you in practice.
Cheers,
Sebastian