I think in the toy model description, the fact of considering the electrons as single point charges is important.
The positive monopole shift contribution appears due to the overlap of the electrons inside the nuclei, which means that part of the negative electric charge can be found inside the nuclei. In the toy model case, we are comparing different systems. In the first one, we have two negative charge and two positive charge. In the second system we have the same as in the first system, with the addition of an other negative charge in the middle of the two positive charges.
In resume: in the second system of the toy model the total negative charge of the system is higher than in the first system of the toy model, so that is why the energy is lower (more coulomb attraction), meanwhile in the overlap calculation, there isn’t an increase of the net negative charge in the system, but the possibility of finding part of these charge inside the nuclei.