The notion you are looking for is well-known in homotopy theory under the name Reedy cofibration, but for some reason this name doesn't show up in papers about Waldhausen categories, even though the concept is used all the time.
To keep things close to your question let's say that $J$ is a finite poset (in general it can be any direct category). For a diagram $X : J \to \mathcal{C}$ and $j \in J$ we define a latching object $L_j X$ as the colimit of the restriction of $X$ to the subposet $\lbrace i \in J \mid i < j \rbrace$. If $L_j X$ exists, then it comes with a canonical map $L_j X \to X_j$. We say that a diagram $X$ is Reedy cofibrant if all $L_j X$s exist and the canonical maps $L_j X \to X_j$ are cofibrations. More generally, a map of Reedy cofibrant diagrams $X \to Y$ is a Reedy cofibration if all the induced maps $X_j \sqcup_{L_j X} L_j Y \to Y_j$ are cofibrations.
You can easily verify that if $\mathcal{C}$ is a category with cofibrations, then so is the category of Reedy cofibrant diagrams $J \to \mathcal{C}$ (and Reedy cofibrations as cofibrations). The same holds for Waldhausen categories.
Now $F_n \mathcal{C}$ is nothing else, but the category of Reedy cofibrant diagrams $[n] \to \mathcal{C}$ and you can easily verify that both categories $F_n F_m \mathcal{C}$ and $F_m F_n \mathcal{C}$ can be identified with the category of Reedy cofibrant diagrams $[m] \times [n] \to \mathcal{C}$ (as categories with cofibrations), which fills in the gap in the proof.