Is the Hausdorff metric on sub-$\sigma$-fields separable?

Take a sequence $A_n$ of independent sets of measure $1/2$. Given two different subsets $B$ and $C$ of natural numbers, suppose WLOG that there is an $n$ in $B\sim C$. Now $\mu(A_n\Delta A) = 1/2$ for all sets $A$ which are independent of $A_n$, so the distance from the sigma algebra generated by $(A_n)_{n\in B}$ to the sigma algebra generated by $(A_n)_{n\in C}$ is at least $1/2$. This shows that the density character of your space is at least the continuum.