Actually, one doesn't need the comparison lemma in this case. As it turns out, $\mathbf{Man}$ is the Karoubi envelope of $\mathbf{Open},$ (see the Examples section of http://ncatlab.org/nlab/show/Karoubi+envelope), which implies that if $$i:\mathbf{Open} \hookrightarrow \mathbf{Man}$$ is the canonical inclusion, the induced restriction functor $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ between their categories of presheaves of $n$-groupoids for any $n$ is already an equivalence.
Edit: In this question (Proof that the category of presheaves on a category $C$ is equivalent to the category of presheaves on its Karoubi envelope) it discusses that presheaves of sets on the Karoubi envelope of a small category is equivalent to presheaves on the original category, and gives a reference.
Now, consider the functor $i_!:Psh_\infty\left(\mathbf{Open} \right) \to Psh_\infty\left(\mathbf{Man}\right)$ which is left adjoint to $i^*.$ Consider the composite $i^*i_!$ which is colimit preserving. It also restricts to an equivalence on $0$-truncated objects, by the above. If $F$ is an arbitrary presheaf on $\mathbf{Open}$, then $F$ can be represented as a simplicial presheaf, hence there exists a simplicial diagram $c_F:\Delta^{op} \to Psh_\infty\left(\mathbf{Open} \right)$ for which each ${c_F}_n$ is $0$-truncated and such that the colimit of $c_F$ is $F.$ Since $i^*i_!$ is colimit preserving, it must send $F$ to itself. A similary argument works using the composite $i_!i^*$, and one concludes that $i_!$ and $i^*$ form an equivalence of $\infty$-categories. In particular, they restrict to an equivalence between $n$-truncated objects for any $n$, hence the induced map $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is an equivalence for all $n$.