If we didnt do first a multipole expansion, the problem would become too complex to handle. The nucleus-electron interaction wouldnt have a simple form, which would make it difficult to calculate small corrections. It would be really difficult because monopole expansion allows us to break the interaction into monopole, quadrupole, and without it we wouldnt know which contributions matter more and the corrections we have to account for would propoably be too big for perturbation theory to work properly.