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Maybe the apparent contradiction can be solved by actually making a multipole expansion for this model. Perhaps the monopole shift is still positive, but the higher order corrections together lead to the overall negative contribution and lowering of the energy.
Otherwise, maybe it has to do with the shape of the nucleus: in the model there are only 2 point charges at a fixed distance from each other instead of a true volume of nuclear charge.
After looking through some of the submissions of other students, I agree with the idea that the apparent contradiction can be solved by realising that the model is no longer electrically neutral upon adding the additional negative charge at the centre. An atom, on the other hand, of course is electrically neutral.
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