It is instructive to consider a simpler case, namely that of chain complexes over a ring. Hovey deals with this example in great detail in his book *Model Categories.* In particular, on page 114 Hovey states that $Ch(R)$ is not a simplicial model category. In algebraic situations like this, it's not really the right question to ask a model category to be simplicial. Instead, one should ask it to be a dg-model category, a.k.a. a $Ch(\mathbb{Z})$-model category. This does hold for $Ch(R)$ with the injective model structure, and should probably hold in the generality of your question as well, though I haven't checked it. A good reference is chapter 4 of Hovey's book, where he defines the notion of a $\mathscr{C}$-model category $\mathscr{M}$ where $\mathscr{C}$ is a model category. This is a generalization of the notion of a simplicial model category (when $\mathscr{C}$ is simplicial sets), i.e. the coherence between $\mathscr{M}$ and $\mathscr{C}$ is analogous to Quillen's SM7 axiom, though Hovey was the first to realize the necessity of the unit axiom for general $\mathscr{C}$.

Note that the model categories $Ch(A)$ in your question are certainly Quillen equivalent to simplicial model categories, e.g. by Dan Dugger's work (because they are combinatorial model categories). Dugger's paper is called "Combinatorial model categories have presentations"

EDIT: Because you're in an additive setting, you actually have more machinery at your disposal than I realized. If you wish to compare an enrichment over $\mathscr{C}$ with an enrichment over $\mathscr{D}$ (e.g. comparing dg-model categories with simplicial model categories), then a great resource is the paper *Enriched Model Categories* by Dugger and Shipley, especially their notion of Quillen adjoint module. I only just found this paper today. Their machinery can be used to say when two enrichments are quasi-equivalent, i.e. equivalent up to homotopy. Now, monoids in sSet and Ch(k) are not so different (thanks to Dold-Kan) so you may be able to use this machinery to beef up the answer I gave from "Quillen equivalent to a simplicial model category" to something stronger. For example, their Proposition 5.2 and their section 9 let you get your hands on what's going on a bit more than the fully general results of Dugger I mentioned previously. On the other hand, if what I wrote before is enough for whatever application you had in mind then feel free to ignore this.