I recently thought about this problem for a general model category rather than $sSet$ and for a different tensor product. I'll tell you what I came up with, because maybe the references can help you, and because I want to write it down somewhere anyway. So let's suppose $A$ is a small category and $M$ is a cofibrantly generated monoidal model category. Define a product on $Fun(A,M)$ by $(X\otimes Y)_a = X_a \otimes Y_a$ for $i\in A$.
Fact: If $A$ has finite coproducts then $Fun(A,M)$ satisfies the pushout product axiom. See Sinan Yalin's 2012 paper for a proof. I tried unsuccessfully to prove the converse, but doing so showed the hypothesis about finite coproducts is there to account for the difference between levelwise cofibrations and projective cofibrations in $Fun(A,M)$. So I believe it's necessary.
Fact: If $A$ is a Reedy category then the Reedy model structure on $Fun(A,M)$ satisfies the pushout product axiom. See Barwick Lemma 4.2
If $M$ is combinatorial, we can define the injective model structure on $Fun(A,M)$. Since the cofibrations are defined to be levelwise cofibrations, we get the pushout product axiom here for free. So this covers all the big model structures on diagram categories, for the objectwise tensor product.
As a side note, if $A = \bullet \to \bullet$ then $Fun(A,M)$ is the arrow category and can be given a different monoidal product, namely the box product $f \Box g$ which is used in the pushout product axiom. In work which is soon to appear, Hovey proves $Fun(A,M)$ inherits the pushout product axiom and monoid axiom from $M$.
Of course, none of this addressed the convolution product, which I've never worked with. But hopefully it'll help you, or at least give you some food for thought. I wonder what happens with the Day product on a Reedy model category, for example.