My question

The background/notation for all of the content of this post is in Lurie, Higher Algebra [HA], Ch. 1.2.3. Everything is purely 1-categorical. Let $\mathcal{A}$ a semiadditive category (with biproducts and 0). Then there is a Dold-Kan functor $$ \mathrm{DK}\colon\mathrm{Ch}_{\geq0}(\mathcal{A})\to\mathbf{s}\mathcal{A}, $$ where $\mathbf{s}\mathcal{A}=[\mathbf{\Delta}^{\mathrm{op}},\mathcal{A}]$ and $\mathrm{Ch}_{\geq0}(\mathcal{A})$ is non-negatively graded chain complexes in objects of $\mathcal{A}$. Take $\mathcal{A}=\mathbf{CMon}$ the category of commutative monoids. My question is whether $\mathrm{DK}$ fully faithful in this case.

Brief background

The definition of the functor is in [HA, Construction 1.2.3.5]; it is defined as a composite $\mathrm{Ch}_{\geq0}(\mathcal{A})\to[\mathbf{\Delta}_{inj},\mathcal{A}]\to\mathbf{s}\mathcal{A}$ with the middle category begin semisimplicial $\mathcal{A}$-objects. The first functor takes a chain complex $(A_*,\partial)$ to the semisimplicial object $[n]\mapsto A_n$ and on all non-identity order preserving injections it is zero except for $d^n$, which is sent to $\partial_n$. The second functor sends a semisimplicial object $A_\bullet$ to the sum $$ [n]\mapsto\bigoplus_{\phi\colon[n]\twoheadrightarrow[k]}A_k $$ indexed by surjections in $\mathbf{\Delta}$. On maps $[n]\to[n']$ it defined using the unique factorization into a split epi + mono - the details are in [HA, Construction 1.2.3.5].

If we take $\mathcal{A}=\mathbf{Ab}$, then $\mathrm{DK}$ is an equivalence by the Dold-Kan correspondence; more generally the same is true if $\mathcal{A}$ is any additive idempotent-complete category [HA, Thm. 1.2.3.7]. If $\mathcal{A}$ is only additive, then $\mathrm{DK}$ is only fully faithful. This can be easily deduced, as Lurie does, from standard Dold-Kan.

However, visibly $\mathrm{DK}$ makes sense for any semiadditive $\mathcal{A}$. I am interested in the case of $\mathbf{CMon}$ as posed in the question because I think that using a purely categorical argument this will show $\mathrm{DK}$ is fully faithful for any semiadditive category.

What I have so far...

1. $\mathrm{DK}$ is faithful. This is clear from the definition since for each $n$ we can recover $f_n$ from $\mathrm{DK}(f)_n$ via the biproduct maps.
2. If $g\colon\mathrm{DK}(A_*)\to\mathrm{DK}(B_*)$ is a simplicial map, using the argument in (1) we can recover the unique $f$ which we hope satisfies $\mathrm{DK}(f)=g$. Because $\mathrm{DK}$ is defined using the boundary map, I was able to show that this $f$ is indeed a chain map by writing out some diagrams.
3. I think, from this, it suffices to show that each $g_n\colon\mathrm{DK}(A)_n\to\mathrm{DK}(B)_n$ is given by a "diagonal matrix" in the sense of the defining direct sum.

As it stands, I don't have a strong guess as to whether this is true or not. Unfortunately the original Dold-Kan proof doesn't help since we cannot define a chain complex from a simplicial commutative monoid - the standard construction needs $-1$.

If this isn't true, a counterexample in $\mathbf{CMon}$ would of course be perfect, but by my earlier remark a counterexample for any semiadditive category would suffice (e.g. maybe there's some obvious reason chain complexes in some $\mathcal{A}$ can't embed into simplicial objects). But this would have to be a non-additive semiadditive category...

Any ideas are appreciated!