Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can consider the adjunction:

$$ F: C \leftrightarrows \mathcal{O}\text{-}alg(C): U $$

between $C$ and the category of $\mathcal{O}$-algebras.

Now if $C$ is a Quillen model category, we can ask whether we can transfer this model structure along this adjunction to get a model structure on $\mathcal{O}$-algebras where a map is a weak equivalence or fibration if it is so after forgetting back to $C$.

There are various known conditions on $\mathcal{O}$ and $C$ for when this is possible. Most of them are fairly mild conditions (like $C$ is cofibrantly generated, $\mathcal{O}$ satisfies some cofibrancy condition). However every treatment I have been able to find starts with a strong assumption that $C$ is a *monoidal model category*. This means there is a strong compatibility between the monoidal structure on $C$ and the model category sptructure. Specifically the pushout-product axiom is satisfied.

Now this is all well and good. After all, it seems reasonable to expect that in order to get a good model structure on $\mathcal{O}$-algebras you would have to start with a model structure which plays well with the monoidal structure on $C$. However I find myself in the situation where I would still like some sort of transfer result like this, but where $C$ is decidedly not a monoidal model category in this strong sense.

In my specific case, $C$ is a Bousfield localization of the model category of simplicial presheaves on a nice Reedy category (with, say the injective model structure). So $C$ is as nice as it could be: it is combinatorial (so cofibrantly generated), the cofibrations are the monomorphisms, in this case it is a simplicial model category, and all the domains of the gen. cofib/gen. acyclic cofibrations are cofibrant, etc.

The operad I want to use is (a simplicial version of) the $E_\infty$-operad, so satisfies very nice cofibrancy conditions.

The monoidal structure on $C$ is the Cartesian product. The terminal object (unit of this monoidal structure) is fibrant and cofibrant. However $C$ is not a Cartesian model category and it definitely fails to satisfy the pushout-product axiom.

So my question is:

Are there any general or specific conditions that would allow one to construct a reasonable model structure on the category $\mathcal{O}\text{-}alg(C)$, even when $C$ fails the pushout-product axiom? Specifically are there conditions which might apply to the case I vaguely described above?