what do you mean by NLO response? effective NLO coefficient? I don't think hiperpolarizabilities have any relationship with transparency, since the latter is connected with bandgap. Prof. Zheshuai Lin in our institute have been doing simulations of NLO properties. Contact him.

Molecular hyperpolarizabilities and, in turn, crystal nonlinear susceptibilities can be defined by a number of constans, matrix elements of the dipole moment operator, frequencies of the transitions and pump frequency. See, e.g.

Chemla D.S., Zyss J. (Eds.), Nonlinear optical properties of organic molecules and crystals, Academic Press, New York, 1987

 For SHG case, the differences of the frequency squares are in the denominators and the product of dipole matrix elements are in the numerators  of so called one- and two-state  terms.  Then, the greater matrix elements (and the smaller the differences of the frequency squares) the greater the hyperpolarizability.  In the absorption spectrum both transition frequency and off-diagonal dopole matrix element define oscillator strength. Transition frequency, converted to the energy  is usually considered as the energy of the peak of the respective absorption band.  So, to obtain resonance increase of the nonlinrearity it is necessary use a molecule with one of it's transitions close to SH photon energy. Hope, this will be useful.

Exists the  known Miller' rule conserning the nonlinear optical response magnitude: the more is linear index refraction so the more to be expected nonlinear optical one. Transparency is determined by absorption, light scattering and reflection. Therefore the less is  transparancy the more strong nonlinear responce for given frequancy is expected. But exist  important details- symmetry of crystall which can forbid some transitions and some  nonlinear coefficient. This details You can  find in books devoted to nonlinear optics.

If the linear absorption of a material is more it will generate heat radiation and on incidence of high intensity laser radiation damage will occur in the material. Therefore, in any nonlinear optical experiments it is desirable to chose a NLO crystal which have low linear absorption i.e. the crystal should be transparent in the wavelength range of interests.