So I am not aware whether you have regarded the previous answer properly or you know about this, but there is a uniqueness theorem for the solution of the <PERSON> equation. Thus this solution is already correct because you have solved the boundary problem:
You have found a solution for the <PERSON> equation in the area where no free charges exist ($\mathbb{R}^3\backslash{0}$), namely seperately for the cavity $\Omega$ and its outside $\mathbb{R}^3\backslash{\Omega}$. You can superpose the dipole potential upon this, since the Poisson/Laplace equation are linear. The boundary here is the border of the cavity $\partial\Omega$ and you impose <PERSON> boundary conditions according to the behaviour of the electric (displacement) field on the boundaries of dielectrics. You can then formally find the coefficients by using the orthonormality of the Legendre polynomials, which is the function that is the general solution to the radially symmetric Laplace equation (as already stated in the solution). With the uniqueness theorem, the problem is solved.
So your objection appears to boil down to the fact that you doubt that the external field and polarisation are not being taken into account at the center of the cavity. Thus I might add 2 remarks:
1. 976
When you're at the origin, the external potential doesn't matter since the potential diverges anyway. In a way the origin is an irregular, singular point. Solving the <PERSON> equation rigorously yields an additional delta at the origin, and being an irregular distribution this really makes the solution of this equation miss out the origin, strictly speaking.
2. Trying to explain the field component by arguing over collective polarisation components seems unreasonable to me. The dipole-dipole interaction is not really trivial, every induced dipole can and will contribute to other dipoles and finally the central dipole also influences the induced charges. Also mind the form of a dipole field which is not radially symmetric but also has a polar angular dependency.
Due to these points, I do not think that it is helpful to depart from just solving the boundary condition problem. Also note the gauge freedom of potential, the electric field is only determined by the spatial variation of the potential (static case). 976