The bi-functional for the non-electrostatic part of the exact embedding potential of frozen-density embedding theory (FDET) depends on whether the embedded part is described by means of a real interacting many-electron system or the reference system of non-interacting electrons (see [Wesolowski, Phys. Rev. A. 77, 11444 (2008)]). The difference δΔFMD[ρA] / δρA(r), where ΔFMD[ρA] is the functional bound from below by the correlation functional Ec[ρA] and from above by zero. Taking into account ΔFMD[ρA] in both the embedding potential and in energy is indispensable for assuring that all calculated quantities are self-consistent and that FDET leads to the exact energy and density in the limit of exact functionals. Since not much is known about good approximations for ΔFMD[ρA], we examine numerically the adequacy of neglecting ΔFMD[ρA] entirely. To this end, we analyze the significance of δΔFMD[ρA] / δρA(r) in the case where the magnitude of ΔFMD[ρA] is the largest, i.e., for Hartree-Fock wavefunction. In hydrogen bonded model systems, neglecting δΔFMD[ρA] / δρ(r) in the embedding potential marginally affects the total energy (less than 5% change in the interaction energy) but results in qualitative changes in the calculated hydrogen-bonding induced shifts of the orbital energies. Based on this estimation, we conclude that neglecting δΔFMD[ρA] / δρA(r) may represent a good approximation for multi-reference variational methods using adequate choice for the active space. Doing the same for single-reference perturbative methods is not recommended. Not only it leads to violation of self-consistency but might result in large effect on orbital energies. It is shown also that the errors in total energy due to neglecting δΔFMD[ρA] / δρA(r) do not cancel but rather add up to the errors due to approximation for the bi-functional of the non-additive kinetic potential.