Pushouts of commutative pseudomonoids

To summarize some of the comments:

I don't know a short answer for why a bicategorical coequalizer doesn't work. If you try to give the bicategorical coequalizer the structure and universal property, you'll find that it just doesn't work somewhere. The intuition is that in higher categories, when you have more coherence information, you generally have to use it rather than ignore it.

The fact that the codescent object is also a codescent object in commutative monoids follows from the fact that it is a reflexive codescent object, and that a two-variable functor preserving reflexive codescent objects in each variable separately also preserves them in both variables jointly. This categorifies the corresponding fact for reflexive coequalizers in 1-categories, and decategorifies a corresponding statement for geometric realizations of simplicial objects in $\infty$-categories; your question here was answered with a proof.

Finally, in $\rm Cat$ (or other locally presentable 2-categories) one can alternatively use the technology of Blackwell-Kelly-Power "Two-dimensional monad theory" to construct colimits in categories of (commutative) monoids, since they are of the form $T\rm Alg$ for an accessible 2-monad $T$.