An Abbé refractometer or a critical angle refractometer (e.g. VCH-1 device) could be used but this only measures the refractive index of the very superficial layer.
Holographic interferometry is a good approach for measuring refractive index variations in inhomogeneous phase objects. My group learned how to do this from Gorski and Kujawinska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Optics in Lasers and Engineering 38, 373-385 (2002).
You can try to use the numerical methods such as " Infinite element" by choasing an arbitrary range of solution the determine the correct on by the way...