In this study, we present a modified regularized lattice Boltzmann method (RLBM) designed to simulate double-diffusive convection of non-Newtonian fluids within heterogeneously porous media. The modification involves the incorporation of correction terms related to shear rate and heat (or concentration) flux into the evolution equations of the RLBM for hydrodynamic equations and convection-diffusion equation, respectively. This allows shear-dependent viscosities and diffusion coefficients to be controlled by additional parameters, enabling the relaxation coefficients in the collision process to remain fixed at optimal values. Through multi-scale Chapman-Enskog analysis, the modified RLBM accurately recovers the governing equations for double-diffusive convection of non-Newtonian fluids within porous media at the representative elementary volume scale. The validity of the method is demonstrated by simulating double-diffusive natural convection in a two-dimensional porous cavity filled with power-law and viscoplastic fluids, with the numerical results showing strong agreement with established data from prior studies. Additionally, the influences of key dimensionless parameters, such as porosity, Bingham number, and power-law index, are thoroughly examined across a range of values.