Yes, the tensor product with a cofibrant replacement can be turned into a lax monoidal functor, where the lax structure maps are weak equivalences.

Consider the model category $\def\Mon{{\rm Mon}} \Mon(M)$ of monoids in $M$. This model structure exists if $M$ satisfies the monoid axiom, which is almost always true in practice.

Consider a cofibrant replacement $Q$ of the monoid $1$ in $\Mon(M)$. The underlying object of $Q$ is a cofibrant object of $\Mon(M)$. (See, for example, Theorem 6.7 in arXiv:1410.5675, but earlier references probably exist.)

Now, the monoid structure of $Q$ equips the functor $Q⊗-$ with a structure of a lax monoidal functor whose lax structure maps are weak equivalences because of the unit axiom for $C$.

Yes, the tensor product with a cofibrant replacement can be turned into a lax monoidal functor, where the lax structure maps are weak equivalences.

Consider the model category $\def\Mon{{\rm Mon}} \Mon(M)$ of monoids in $M$. This model structure exists if $M$ satisfies the monoid axiom, which is almost always true in practice.

Consider a cofibrant replacement $Q$ of the monoid $1$ in $\Mon(M)$. The underlying object of $Q$ is a cofibrant object in $M$. (See, for example, Theorem 6.7 in arXiv:1410.5675, but earlier references probably exist.)

Now, the monoid structure of $Q$ equips the functor $Q⊗-$ with a structure of a lax monoidal functor whose lax structure maps are weak equivalences because of the unit axiom for $C$.