It's not necessary to make use of various different tests to determine the material parameters of a hyperelasticity relation. Uniaxial tensile tests (under the assumption of incompressibility or the assumption of a bulk modulus) are in a first step sufficient. The question is more whether the data shows also viscoelastic effects (then hyperelasticity cannot represent the material behavior), or if the model can predict three-dimensional material in an appropriate manner. In some situations the three-dimensional behavior is better predicted if in the parameter identification process more information of the material behavior is considered. In this respect, I refer to

Haupt, P., Sedlan, K.: Viscoplasticity of elastomeric materials. Experimental facts and constitutive modeling. Archive of Applied Mechanics 71 (2001), 89-109

and the literature cited therein.

The number of material constants depends on the model used. A neo-Hookean model, among others, entails one constant. A Mooney-Rivlin Model requires 2. And other models may require 3 or more. So you must know the model under consideration.

Run several kinds of tests for the data to be studied. Are the data the same for various directions of deformation? What can be said of the isotropic character of the material?  Is the material "incompressible"? What is the Poisson Ratio at various stages of tensile loading. Can you start with the general model for an isotropic material?. See some of my papers on the Poisson Function and other topics, and especially the early experiments of Rivlin and Saunders in the 1940s. You need to do some literature research for yourself to learn what others have done. Best wishes for your studies.