Is this an Error in Griffiths Electrodynamics?

Check Problem 3.43 in Griffiths Introduction to Electrodynamics

A conducting sphere of radius $a$, at potential $V_0$, is surrounded by a thin concentric spherical shell of radius $b$, over which someone has glued a surface charge $sigma(theta)=kcos(theta)$ where $k$ is a constant and $theta$ is the polar angle.

It then asks to find the potential in the $r>b$ and $a<r<b$. The answer provided by the book is:
$V(r, θ) = frac{aV_0}{r} + frac{(b^3 − a^3)k cos θ}{3r^2 epsilon_0}, r ≥ b,$

$V(r, θ) = frac{aV_0}{r} + frac{(r^3 − a^3)k cos θ}{3r^2 epsilon_0}, r ≤ b$

Credit: Griffiths, David J.. Introduction to Electrodynamics (p. 162). Cambridge University Press. Kindle Edition.

Solving for the region in between the discs: using the boundary condition $V(a,theta)=V_0$ we find that:
$$V(a,theta)=sum_{l=0}^{infty}{(A_la^{l}+frac{B_l}{a^{l+1}})P_l(costheta)}=V_0$$
Since $V_0$ is a constant, and thus has no $theta$ dependence, we conclude the only term in the series must be the one with $l=0$ to ensure that the left side of the equation has no terms in $cos theta$. Thus in summary we find that:
$$A_0+frac{B_0}{a}=V_0$$
and thus the potential has the form:
$$V(r,theta)=A_0+frac{a(V_0-A_0)}{r}$$
but this will obviously not satisfy the form given by the answer in the book, as it must have a $cos theta$ term and of course the $A_l$‘s and $B_l$‘s are constants. I am probably at fault, but I don’t see where exactly. Please help.