Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the category of commutative pseudomonoids $\mathrm{CMon}(\mathcal{C})$ has bicategorical pushouts.

This is true for symmetric monoidal categories $(\mathcal{C},\otimes)$. Here one constructs the pushout of monoid morphisms $(A,\mu_A,\eta_A) \leftarrow (C,\mu_C,\eta_C) \rightarrow (B,\mu_B,\eta_B)$ as the coequalizer $A \otimes_C B$ of the two evident morphisms $A \otimes C \otimes B \rightrightarrows A \otimes B$ in $\mathcal{C}$ and endows it with a monoid structure. The most familiar case for this is $\mathcal{C}=\mathsf{Ab}$, which yields the construction of pushouts of commutative rings.

I don't see why this should not work for symmetric monoidal bicategories. But I was told that one has to use codescent objects; I wonder why this is necessary. A reference for coproducts in $\mathrm{CMon}(\mathcal{C})$ (which is the special case $C=\mathbf{1}_{\mathcal{C}}$) is Theorem 5.2 in Schäppi's paper on ind-abelian categories.

Edit: I think I now understand why coequalizers are not enough. So assume that $\mathcal{C}$ has bicategorical codescent objects which are preserved by $\otimes$ in each variable (add additional assumptions if necessary). Does then $\mathrm{CMon}(\mathcal{C})$ have bicategorical pushouts? It seems that Schäppi uses this (in the special case $\mathcal{C}=\mathsf{cat}_{\mathrm{fc}/k}$, the category of essentially small finitely cocomplete $k$-linear categories) without proof here in Section 4.1. I assume that the codescent object is constructed in $\mathcal{C}$ (since we cannot just assume the existence of codescent objects in $\mathrm{CMon}(\mathcal{C}$), right?), but the proof that it has the structure of a commutative pseudomonoid, and that it is actually a codescent object in $\mathrm{CMon}(\mathcal{C})$, is missing.

I would already be happy for a detailed explanation or reference for the case $\mathcal{C}=\mathsf{cat}$, i.e. how to construct bicategorical pushouts of small symmetric monoidal categories.