$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $M$, etc: in particular, assume $\C$ has all colimits and the zero object $\emptyset$ is preserved by $M$.
I'm interested in a model for the "free product" $A*B: = A\sqcup_\emptyset B$ (and more generally, in colimits of simple diagrams of algebras).
Under the additional assumption that everything is well-behaved with respect to some model structures and the monad is particularly nice, Batanin and Berger's model structure on algebras over polynomial monads gives a model for $A*B$ assuming some cofibrantness conditions. I'm interested in cases where this formalism doesn't quite work.
It seems to me that there should be a "bar resolution" model defined without reference to a model structure, with similar flavor to Riehl's weighted limit formalism. Namely, if one considers the set of objects $$M^l(M^m(A)\sqcup M^n(B))$$ (for $l,m,n\ge 0$) these have natural maps between them indexed by a certain category of "Y-shaped" posets (with legs of the Y having length $l,m,n$) and it should follow from formal nonsense that the colimit of this diagram in $\C$ should be an $M$-algebra which computes the colimt in nice cases.
Is this true, and if it is then does there exist a there a reference for this, or does it follow from some more general infinity-categorical formalism?
