Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious enriched full embedding of the subcategory of fibrant objects $\mathbf{S}^\circ \hookrightarrow \mathbf{S}$. 


Since $\mathbf{S}$ is a combinatorial model category, the small object argument provides us with a fibrant replacement functor $\mathbf{S}\to \mathbf{S}^\circ$ that is *not necessarily enriched*. 


First question: If $\mathcal{A}$ is an $\mathbf{S}$-enriched combinatorial model category (maybe in which all objects are cofibrant), is there an enriched version of the small object argument that provides us with a fibrant replacement functor $\mathcal{A}\to \mathcal{A}^\circ$ that is an *enriched* functor?


Second question: If the first question is not true in general for $\mathbf{S}$-enriched combinatorial model categories, is it at least true for the special case $\mathcal{A}=\mathbf{S}$ itself? 



Excellent monoidal model categories admit enriched fibrant replacement functors?
I think the answer is yes (to both questions). In Emily Riehl's book "Categorical homotopy theory", chapter 13 is all about the enriched small object argument. Theorem 13.2.1 on page 177 (I hope I'm looking at a version close to the final one) explains when the enriched small object argument works. A condition Riehl calls $(\star)$ is required. She remarks (13.2.3) that this condition is satisfied if, for every $X$ in $\mathcal{A}$, $-\otimes X$ preserves cofibrations and trivial cofibrations. Clearly, this is satisfied in your setting. If you define your fibrant replacement functor via the enriched weak factorization system (factoring $X\to \ast$ into a trivial cofibration $X\to RX$ and then a fibration $RX\to \ast$), then the functor $R$ is $\mathcal{S}$-enriched. This is the content of Riehl's Corollary 13.2.4.


