Maximum-entropy-inspired interpolative-based moment closure methods are considered for the prediction of nonequilibrium radiative transfer in non-gray participating media, including application to sooting laminar flames. In particular, interpolative-based versions of both the first- and second-order maximum entropy closures, M1 and M2, respectively, are described and examined. The interpolative-based closures are constructed so as to duplicate many of the desirable mathematical properties of maximum-entropy closure techniques, including positivity and moment realizablility of the distribution of radiative energy and hyperbolicity of the resulting moment equations while offering significant computational savings compared to approaches that involve the direct numerical solution of the optimization problem for entropy maximization. Theoretical details of the proposed interpolative-based moment closures, along with a description of an efficient Godunov-type finite-volume scheme that has been developed for the numerical solution of the moment equations on multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR) are described. The predictive capabilities of the proposed non-gray interpolative-based M1 and M2 closures are assessed by considering their application to canonical radiative transfer problems involving transport through real gases with a strong spectral dependence of the absorption coefficient as well as to gaseous, methane-air, sooting, laminar, co-flow, diffusion flames at elevated pressures, for which the absorption properties of the gases are evaluated using a statistical narrow-band correlated-k model. The numerical results demonstrate the potential of the maximum-entropy-inspired closures. In particular, the non-gray interpolative-based M1 and M2 closures are shown to provide improved predictions compared to the standard P1 spherical harmonic moment closure and are of comparable or even of improved accuracy compared to the third-order P3 closure, while providing additional computational robustness relative to the spherical harmonic closures.