I think you probably need to prove the following. Let $0 \rightarrow B_0 \xrightarrow{f} B \xrightarrow{g} B_1 \rightarrow 0$ be a SES, and suppose that $B'$ is some module such that $0 \rightarrow B_0 \otimes B' \rightarrow B \otimes B' \rightarrow B_1 \otimes B' \rightarrow 0$ is also a SES. I don't know how one could set this up if we don't assume the tensored sequence isn't exact. We need to show that the following square commutes
where the vertical maps are given by the exterior product. Here is a sketch. Let $P, P', P''$ be projective resolutions of $A, A', A \otimes A'$, respectively. Recall that there is a map $P \otimes P' \rightarrow P''$ that is unique up to chain homotopy equivalence, where $P \otimes P'$ denotes the total complex of the tensor double complex. Our goal is then to show that
commutes. The bottom square commutes by naturality of the connecting homomorphism. To check the commutativity of the top square, we argue directly on the level of chains and use the explicit description of the connecting homomorphism. Let $\alpha = \sum_m p_m \otimes b_m$ and $\beta = \sum_n p_n' \otimes b_n'$ be cycles representing elements of $H_i(P \otimes B_1)$ and $H_j(P' \otimes B')$, respectively. On one hand, we have
$$
(\delta \otimes \mathrm{id})([\alpha]\otimes[\beta]) = [\delta \alpha] \otimes [\beta] \mapsto [(\mathrm{id} \otimes f)^{-1}(\sum_m dp_m \otimes g^{-1}b_m) \otimes \beta].
$$
On the other hand,
\begin{align*}
\delta[\alpha \otimes \beta] &= \delta[\sum_{m,n} p_m \otimes p_n' \otimes b_m \otimes b_n'] \\
&= [(\mathrm{id} \otimes \mathrm{id} \otimes f \otimes \mathrm{id})^{-1} (\sum_{m,n} dp_m \otimes p_n' \otimes g^{-1}b_m \otimes b_n' - p_m \otimes dp_n' \otimes g^{-1}b_m \otimes b_n')] \\
&= [(\mathrm{id} \otimes f \otimes \mathrm{id} \otimes \mathrm{id})^{-1} ((\sum_m dp_m \otimes g^{-1}b_m) \otimes \beta - (\sum_m p_m \otimes g^{-1}b_m) \otimes d \beta)] \\
&= [(\mathrm{id} \otimes f \otimes \mathrm{id} \otimes \mathrm{id})^{-1} (\sum_m dp_m \otimes g^{-1}b_m) \otimes \beta] \\
&= [(\mathrm{id} \otimes f)^{-1}(\sum_m dp_m \otimes g^{-1}b_m) \otimes \beta]
\end{align*}
where we have used the fact that $d \beta = 0$. Also note that I've been freely commuting the tensors. This proves that the upper square commutes.
