There are elastomers that are elastic out to 300% with no permanent deformation. That is what the Hyperelasticity material models are for.

You have not provided any data to differentiate where plasticity occurs! There is not plasticity at the beginning, right? Even when it begins, some of the stretch will be elastic while some will be plastic. The stress-strain curve alone is not sufficient because both elastic and plastic deformation are blended into the one number.