In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\rightarrow B$ be a fibration with fiber $F$ s.t. $\pi_1(B)$ acts trivially on $F$ (in the homotopical sense), $B$ is a path-connected CW-complex and $h_{\bullet}$ is an additive homology theory satisfying the axiom of weak equivalence. Take the skeletal filtration $B_0\subseteq B_1\subseteq\dotsc\subseteq B$ of $B$ and pull it back to a filtration $E_0\subseteq E_1\subseteq\dotsc\subseteq E$, which induces a convergent right-half plane homological spectral sequence $E_{pq}^1=h_{p+q}(E_p,E_{p-1})\Rightarrow h_{p+q}(E)$ (the differential $d_1$ is given by the boundary operator).
Picking characteristic maps $(\Phi_{\alpha},\varphi_{\alpha})\colon(D^p_{\alpha},S^{p-1}_{\alpha})\rightarrow(B^p,B^{p-1})$ for the $p$-cells of $B$, the fibration trivializes when pulled back along $\Phi_{\alpha}$ to the contractible space $D^p_{\alpha}$, so there is a fiber-homotopy equivalence $\Phi_{\alpha}^{\ast}E_p\simeq D^p_{\alpha}\times F$ and hence an identification $\bigvee_{\alpha}S^p_{\alpha}\wedge F_+\simeq\bigvee_{\alpha}(D^p_{\alpha}\times F)/(S^{p-1}_{\alpha}\times F)\cong E_p/E_{p-1}$. Thus, we have $h_{p+q}(E_p,E_{p-1})\cong\bigoplus_{\alpha}h_q(F)$, the $p$-th cellular chain group of $B$ with coefficients in $h_q(F)$. The salient fact is that the differential $d_1$ corresponds to the differential in the cellular chain complex, so that we can identify the second page as $E_{pq}^2=H_p(B;h_q(F))$.
Now, May claims that this is true and "straightforward". The idea given is that the differentials are induced by a map $E_p/E_{p-1}\rightarrow\Sigma E_{p-1}\rightarrow\Sigma(E_{p-1}/E_{p-2})$, where the first is the coboundary map in the Barratt-Puppe sequence of $E_{p-1}\hookrightarrow E_p$ (perhaps up to sign) and the latter is induced by projection. The claim would be implied if this sequence, up to homotopy, can be obtained from the corresponding sequence in the base by smashing with $F_+$, but it is not clear to me if that is the case.
Question: How can this argument be completed?
The only other place in the literature where I could find an argument in this spirit (I am aware of arguments in different spirits) is in Jeffrey Strom's Modern Classical Homotopy Theory, Chapter 28, which unfortunately rests on the book's incorrect Lemma 4.32.
