Monoidal structures on modules over derived coalgebras

Not quite sure what your exact question is, but the general pattern is as follows: let $Pr$ be the $(\infty,1)$-symmetric monoidal category of presentable categories, cocontinuous functors, natural isos between them and so on. Let $S\in E_\infty-alg(Pr)$ be a presnetable symmetric monoidal category. Then you have a symmetric monoidal functors $$E_1-alg(S)\longrightarrow Pr$$ sending $A$ to $A-mod$ and a morphism $A\rightarrow B$ to the corresponding induction functor $-\otimes_A B$. Likewise you have a symmetric monoidal functor $$E_1-coalg(S)\longrightarrow Pr$$ sending $C$ to $C-comod$ and a morphism $C\rightarrow D$ to corestriction. Therefore you get a functor $$E_1-bialg(S)=E_1-alg(E_1-coalg(S))\longrightarrow E_1-alg(Pr)$$ by applying "comod", hence the category of comodules over a bialgebra is $E_1$, i.e. monoidal. Likewise, modules over a bialgebra should really be regarded as a "comonoidal category", i.e. an $E_1$-coalgebra in $Pr$. It is also monoidal basically because restriction along algebra morphisms is also cocontinuous, i.e. there is also a contravariant functor from $E_1-alg(S)$ to $Pr$, but this is somewhat less natural and leads to some techincal issues (already in the classical/non-derived case).

Now applying Dunn's theorem you get similar statements for the higer versions of bialgebras.