It was recently shown (Phys. Rev. Lett. 1994, 72, 3831) that the "test-particle-self-consistent" fundamental measure free energy density functional for general inhomogeneous hard-sphere fluids (J. Chem. Phys. 1993, 98, 8126) predicts phase separation in bulk binary mixtures with large size ratios, R(2)/R(1) > similar to 4, when the packing fractions for the two species are comparable, in qualitative agreement with recent experiments on "nearly hard-sphere" colloidal particles. These preliminary calculations are focused on the spinodals and not on the equilibrium coexistence curve, which is much harder to calculate. To prepare the necessary tools for undertaking the complete thermodynamic calculation, the present paper provides a full description of this self-consistent density functional for mixtures and clears many fine points in the theory. It also gives a more detailed presentation of the results relevant for phase separation. These results should be useful for studying external field effects (e.g., sedimentation and confining walls) on the entropically driven demixing transition, as encountered in real experiments on colloids.