A central limit theorem for partitions involving generalised divisor functions

We let $p_f(n,k)$ count the $f$-restricted coloured partitions of $n$ with $k$ parts, for a positive integer valued function $f$, given by the bivariate generating series \[ Q_f(z,u) := 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{Δ_f(k)}, \] where $Δ_f(n)=