Monoidal model category structure on a functor category.

[This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.]

The result you ask for is a consequence of proposition 2.2.15 in Sam Isaacson's Ph.D. thesis (Harvard University, 2009). That proposition is stated for combinatorial symmetric monoidal (closed) model categories, and not just for simplicial sets. In this general case, the statement in Sam Isaacson's thesis requires the base category to have virtually cofibrant morphism objects. This condition is automatic for simplicial sets, as all simplicial sets are cofibrant in the usual model structure.

I have no idea who first proved or published the above result.