  We study big Hankel operators $H_f^\nu:A^p_\omega \to L^q_\nu$ generated by
radial Bekoll\'e-Bonami weights $\nu$, when $1<p\leq q<\infty$. Here the radial
weight $\omega$ is assumed to satisfy a two-sided doubling condition, and
$A^p_\omega$ denotes the corresponding weighted Bergman space. A
characterization for simultaneous boundedness of $H_f^\nu$ and
$H_{\overline{f}}^\nu$ is provided in terms of a general weighted mean
oscillation. Compared to the case of standard weights that was recently
obtained by Pau, Zhao and Zhu (Indiana Univ. Math. J. 2016), the respective
spaces depend on the weights $\omega$ and $\nu$ in an essentially stronger
sense. This makes our analysis deviate from the blueprint of this more
classical setting. As a consequence of our main result, we also study the case
of anti-analytic symbols.
