Homotopy pushout independent of factorization and symmetric in cofibration category

I don't remember how Baues does this exactly, but all facts of this sort follow from the Gluing Lemma (see Lemma 1.4.1 in this paper) and "Brown type factorization". By this I mean the following construction. Given a morphism $X \to Y$ and two factorizations $X \to Z_0 \to Y$ and $X \to Z_1 \to Y$ (as cofibrations followed by weak equivalences), we factor the induced morphism $Z_0 \sqcup_X Z_1 \to Y$ as $Z_0 \sqcup_X Z_1 \to Z_2 \to Y$ and obtain a factorization $X \to Z_2 \to Y$ that is weakly equivalent to both original ones. If you do that to a map in a span, you will obtain a zig-zag of weak equivalences connecting spans obtained from any two factorizations and you can apply the Gluing Lemma to that. (Note that axiom C4 is never used in this argument.)

(The following was intended as a comment to Karol's answer, that after using the "Brown Type Factorization" trick, we can prove the result by applying a result in the book, which does not assume cofibrantness, but due to the word count, I think it might be more appropriate to present it as an answer.)

First, using the Brown Type Factorization trick, we can reduce to showing the following case : suppose the factorization involving $W,V$ are weakly equivalent, that is, if we have a diagram :

$$\begin{align}B\rightarrow V\rightarrow A\\||\quad\;\downarrow i \;\quad||\\B\rightarrow W\rightarrow A\end{align}$$

(with the middle horizontal arrow being a weak equivalence)

then $W\cup_BD\to C$ is a weak equivalence iff $V\cup_BD\to C$ is a weak equivalence.

By (C1), it then suffices to show $i\cup_B 1_D:V\cup_BD\to W\cup_B D$ is a weak equivalence.

Now the result follows from Lemma 1.2(b) in chapter II in the book applied to the diagram

$$\begin{align}V\leftarrow B\rightarrow D\\i\downarrow\quad\;||\quad\quad||\\W\leftarrow B\rightarrow D\end{align}$$