Let $A$ be a brace algebra and $B$ the Koszul dual bialgebra. There is a natural adjunction
$$
\Omega\colon \mathrm{CoMod}_B\rightleftarrows \mathrm{LMod}_A
$$
where, for instance, the functor $\mathrm{LMod}_A\rightarrow\mathrm{CoMod}_B$ sends $M\mapsto M\otimes B$ equipped with some standard differential.

For $\mathrm{LMod}_A$ the natural notion of weak equivalence is that of quasi-isomorphism and in $\mathrm{CoMod}_B$ a map $C_1\rightarrow C_2$ is a weak equivalence if $\Omega C_1\rightarrow \Omega C_2$ is a quasi-isomorphism. The localization of $\mathrm{LMod}_A$ with respect to quasi-isomorphisms is the derived category of $A$-modules and the localization of $\mathrm{CoMod}_B$ with respect to $\Omega$-quasi-isomorphisms is the coderived category of $B$-comodules. The above adjunction induces an adjoint equivalence between the corresponding localizations (see https://arxiv.org/abs/0905.2621 for details on all of these constructions).

$\mathrm{CoMod}_B$ indeed has a natural monoidal structure. However, $\mathrm{LMod}_A$ does not have any natural monoidal structure (there is no strict way to turn left modules over a brace algebra into right modules). There is, however, a monoidal structure on the derived category of $A$-modules. It is obtained using the Dunn--Lurie additivity equivalence $\mathcal{A}\mathrm{lg}_{\mathbb{E}_2}\cong \mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})$.

Brace bar-cobar duality gives an adjunction $\mathrm{Alg}(\mathrm{CoAlg})\rightleftarrows \mathrm{Alg}_{\mathrm{Brace}}$. After inverting corresponding weak equivalences you obtain another equivalence of $\infty$-categories $\mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})\cong \mathcal{A}\mathrm{lg}_{\mathbb{E}_2}$. As far as I know, nobody has compared Dunn--Lurie additivity for $\mathbb{E}_2$ with brace bar-cobar duality. Assuming such a comparison, the monoidal structure on the coderived category of $B$-comodules will indeed coincide with the monoidal structure on the derived category of $A$-modules.