Not a complete answer.

Let's recall the work of [Dugger and Spivak](https://arxiv.org/abs/0910.0814). Let $\mathcal G$ be a "category of gadgets closed under wedges" (Definition 5.4) (a full subcategory of $sSet_{\ast,\ast}$ satisfying certain conditions). Dugger and Spivak define a functor

$$\mathfrak C^{\mathcal G}: sSet \to sCat$$

$$Ob \mathfrak C^{\mathcal G}(S) = S_0 \qquad \qquad Hom_{\mathfrak C^{\mathcal G}(S)}(a,b) = N(\mathcal G \downarrow S)_{a,b}$$

where $N$ is the ordinary nerve, $\downarrow$ is the ordinary over-category, and the subscript means we restrict to have the correct endpoints. They show (Thm 5.2) that $\mathfrak C^{\mathcal G}$ is connected by a zigzag of natural DK equivalences to the usual functor $\mathfrak C$ (adjoint to the homotopy coherent nerve $\mathcal N$).

Moreover, the map $\theta: \mathfrak C^{\mathcal G}(X) \times \mathfrak C^{\mathcal G}(Y) \to \mathfrak C^{\mathcal G}(Y)$ of Proposition 6.2 appears to make $\mathfrak C^{\mathcal G}$ into a lax monoidal functor! It is defined on homspaces by taking binary products (a category of gadgets is required to be closed under finite products). Be warned -- I have not checked this carefully! If there's a difficulty, then it probably lies in having to choose binary products in a coherent way...

Assuming this is true, the composite functor $R = \mathcal N Ex^\infty_\ast \mathfrak C^{\mathcal G}$ is a composite of lax monoidal functors and so is lax monoidal. It takes values in quasicategories, but unfortunately it is only connected by a zigzag of natural weak equivalences to the identity.

Also, this is all in the setting of the Joyal model structure -- I'm not sure about adapting it to the marked case.