After thinking about this for a little while, it seems like there is a satisfactory way of carrying this out in group cohomology using the Hochschild-Serre spectral sequence of a group extension 
$$1 \to H \to G \to Q \to 1$$ 
(henceforth we will call this the HSSS), and since my original motivating situation happened to be a fibration of aspherical manifolds, this will suffice for my purposes. I will refrain from selecting this as the answer, though, since I would still be curious to know how to carry this out in the topological setting. 

For simplicity, I'll restrict attention to the trivial coefficient module $\mathbb Z$. Suppose that $H$ is a $PD_{m}$-group, $Q$ is a $PD_{n}$-group, and $G$ is a $PD_{m+n}$-group. Chapter VII of Brown's *Cohomology of Groups* contains a construction of the HSSS. In the homological setting, the idea is to show that there is a particular chain complex $C$ for which the homology with coefficients in the chain complex $H_*(Q, C)$ is isomorphic to $H_*(G)$; the HSSS is then derived from one of the filtrations on the bigraded chain complex underlying the definition of $H_*(Q, C)$.

 In fact, letting $F$ be a projective resolution of $\mathbb Z$ over $\mathbb Z G$ of length $m+n$, Brown explains why $C = F_H$ suffices. If we let $F'$ be a projective resolution of $\mathbb Z$ over $\mathbb Z Q$, the upshot is that the tensor product of chain complexes $F' \otimes_Q F_H$ computes $H_*(G)$.

Turning to cohomology, similar arguments show that the homology of the cochain complex
$$
\operatorname{Hom}_Q(F', \operatorname{Hom}_H(F, \mathbb Z))
$$
is given by $H^*(G)$. 

The crucial observation is that for a $PD_k$-group $\Gamma$, if $F$ is a projective resolution of finite length $k$, then $\overline F : = \operatorname{Hom}_\Gamma(F, \mathbb Z \Gamma)$ is also a projective resolution of $\mathbb Z$ over $\mathbb Z \Gamma$. (A technical point which explains how the shift in degrees arises is that one must monkey with the grading on $\overline F$ in the proper way).

We can apply tensor-Hom adjunction to see that
$$
\operatorname{Hom}_Q(F', \operatorname{Hom}_H(F,\mathbb Z)) \cong \overline{F'} \otimes_Q \overline{F}_H.
$$
As $G$ and $Q$ are $PD$-groups by assumption, we may use $\overline{F'}$ and $\overline{F}$ as our projective resolutions in the computation of $H_*(G)$. Paying attention to the bigradings, the $(p,q)$-component of the left-hand side above corresponds to the $(n - p, m-q)$-component on the right.

Now the properties I was looking for above will follow automatically from the spectral sequence machinery.