I am trying to get a contradiction from the following set of hypotheses:

Let $R$ be a ring. Let $M$ be a direct sum of non-zero $R$-modules $M_1$, $M_2$, $\dotsc$. For each $i\ge1$, let $f_i:M_i\to M_{i+1}$ be an $R$-monomorphism that is not onto and let $M'_i=\{x+f_i(x)\mid x\in M_i\}$.

Assume that $M=M_1\oplus M'_2\oplus M_3\oplus M'_4\oplus\dotsb$. For $n\ge1$, let $\psi_{2n}$ be an $R$-isomorphism from $M_{2n}$ onto one of the summands in this direct sum. Assume that $M=M'_1\oplus M'_3\oplus\dotsb\oplus\psi_2(M_2)\oplus \psi_4(M_4)\oplus\dotsb$. Assume that for all $n\ge1$, there exists $m\ge1$ such that $\psi(M_{2m})=M'_{2n}$. Also assume that, for some $n_0\ge1$, for all $n\ge n_0$ we have $\psi_{2n}(M_{2n})=M'_{2m}$ for some $m\ge1$.

Lemma 9 on p. 336 of Harada and Sai, "On categories of indecomposable modules. I," Osaka Math. J. 7 (1970), 323–344 is different than this, but I hope I have extracted what I need above.