I'm not sure what you have ...
a) a stress/strain curve of the material, or
b) a load/deflection curve of the structure (made from this material) ?
If b), and if the material is reasonably homogeneous, then I guess you could build a FEM of the structure, give it some property and adjust until you get the deflections you measured. Mind you, I'd expect more than one solution (for E and G) would work so I don't think it'll tell you something definitive.
another day in paradise, or is paradise one day closer ?