The asymptotic number of weighted partitions with a given number of parts

For a given sequence $b_k$ of non-negative real numbers, the number of weighted partitions of a positive integer $n$ having $m$ parts $c_{n,m}$ has bivariate generating function equal to $\prod_{k=1}^\infty (1-yz^k)^{-b_k}$. Under the assumption that $b_k\sim Ck^{r-1}$, $r>0$, and related conditions on the Dirichlet generating function of the weights $b_k$, we find asymptotics for $c_{n,m}$ when $m=m(n)$ satisfies $m=o\left(n^\frac{r}{r+1}\right)$ and $\lim_{n\to\infty}m/\log^{3+\eps}n=\infty$, $\eps>0$