Let me elaborate on Dmitri Pavlov's excellent answer. (And sorry, Dmitri, for taking this long to digest your answer.)
Let us write $\theta_{\mathrm{proj}}$ for the functor $\theta$ when $\mathbf{A}^\mathcal{C}$ is equipped with the projective model structure, and define $\theta_{\mathrm{inj}}$ similarly. We want to show that $\theta_{\mathrm{inj}}$ is a categorical equivalence.
We start by recalling that if $\mathbf{M}_\Delta$ is a simplicial model category with underlying category $\mathbf{M}$, then the map $(N(\mathbf{M}^\circ),\mathrm{weq})\to (N(\mathbf{M}_\Delta^\circ),\mathrm{weq})$ is a weak equivalence of marked simplicial sets (This is 1.3.4.20 and 1.3.4.16 of Higher Algebra). (In other words, $N(\mathbf{M}^\circ_\Delta)$ is the localization of $N(\mathbf{M}^\circ)$ with respect to weak equiavlences.) Taking $\mathbf{M}_\Delta=\mathbf{A}^{\mathcal{C}}$, we deduce that the map
$$(N((\mathbf{M}^\circ)_{\mathrm{proj}}),\mathrm{weq})\to (N((\mathbf{M}_{\Delta\mathrm{proj}})^\circ, \mathrm{weq})$$
is a weak equivalence of marked simplicial sets. Now let $\mathbf{M}'\subset \mathbf{M}$ denote the full subcategory spanned by the functors taking values in $\mathbf{A}^\circ$. Using the cofibrant replacement functor, we find that the inclusion
$$(N((\mathbf{M}^\circ)_{\mathrm{proj}}),\mathrm{weq})\hookrightarrow (N(\mathbf{M}'),\mathrm{weq})$$ is a weak equivalence of marked simplicial sets. Since $\theta_{\mathrm{proj}}$ is a categorical equiavlence, this means that the map $(N(\mathbf{M}'),\mathrm{weq})\to\operatorname{Fun}(S,N(\mathbf{A}^\circ))^\natural$is a weak equiavlence of marked simplicial sets. This, in turn, implies (by repeating a similar argument) that $\theta_{\mathrm{inj}}$ is a categorical equivalence, and we are done.