  \'O. Blasco and S. Pott showed that the supremum of operator norms over $L^2$
of all bicommutators (with the same symbol) of one-parameter Haar multipliers
dominates the biparameter dyadic product BMO norm of the symbol itself. In the
present work we extend this result to the Bloom setting, and to any exponent
$1<p<\infty$. The main tool is a new characterization in terms of paraproducts
and two-weight John--Nirenberg inequalities for dyadic product BMO in the Bloom
setting. We also extend our results to the whole scale of indexed spaces
between little bmo and product BMO in the general multiparameter setting, with
the appropriate iterated commutator in each case.
