= Solution
For an axisymmetric azimuthal <vector field>, the <curl in spherical coordinates> has components
$$
(\nabla\times\boldsymbol A)_r=\frac1{r\sin\theta}\partial_\theta(\sin\theta A_\phi),\qquad
(\nabla\times\boldsymbol A)_\theta=-\frac1r\partial_r(rA_\phi),\qquad
(\nabla\times\boldsymbol A)_\phi=0.
$$
The angular expressions are $\sin\theta A_\phi^{(+)}=(1-\cos\theta)/r$ and $\sin\theta A_\phi^{(-)}=-(1+\cos\theta)/r$, each with $\theta$ derivative $\sin\theta/r$. Also $rA_\phi$ is independent of $r$, so the second component vanishes. Consequently, on each potential's regular patch,
$$
\boxed{\nabla\times\boldsymbol A^{(+)}=\nabla\times\boldsymbol A^{(-)}=\frac{\boldsymbol e_r}{r^2}=\boldsymbol B.}
$$
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