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Concerning the question what the complications would be if we have a general shape for a nucleus and try and solve the system with perturbation theory without first making a multipole expansion:
The first thing that came to mind is the complexity, i.e. is the problem still solvable (analytically). By making the full multipole expansion and truncating it, an approximate expression can be found which should be easier to solve.
Furthermore, as is shown in the lecture, the multipole expansion provides us with some first order terms that are already solved quantum mechanically. (Nucleus & electron terms)
Lastly, as hinted on slide 6 (around 04:26 in lecture) a distinction in energy scales can be found which show that perturbation theory is indeed applicable since the perturbing energies are reasonably small. (compare meV to eV or even MeV)
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