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.
Another Edit: First I gave the following argument, but it seems to have a gap (skip over this to get to the direct answer):
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 (wait: Why shoudl $i_!$ send $0$-truncated objects to $0$-truncated objects?). 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 $n$.
Unfortunately (see the bold wait) I don't see how to show that $i_!$ preserves $0$-truncated objects (i.e. agrees with the 1-categorical left Kan extension when restricted to presheaves of sets) until I show its an equivalence, so here's another way to finish the argument:
(Start reading again here if you skipped): Consider the restriction functor $$i^*:Psh\left(\mathbf{Man}\right) \to Psh\left(\mathbf{Open} \right)$$ of presheaves of sets, which has a right adjoint $i_*$. (Explicitly, by the Yoneda lemma, one has that $i_*(F)(M)\cong Hom(i^*y(M),F)$ where $y$ is the Yoneda embedding.). Since $i^*$ also has a left adjoint $i_!,$ $i^*$ is left exact, and since $i$ is full and faithful, so is $i_*.$ So we have that the adjunction $(i^*,i_*)$ exhibits $Psh\left(\mathbf{Open} \right)$ as a left exact localization of $Psh\left(\mathbf{Man}\right)$. Hence, there is a unique Grothendieck topology $J$ on $\mathbf{Man}$ for which $Sh_J\left(\mathbf{Man}\right)\cong Psh\left(\mathbf{Open}\right)$ (more precisely, such that the localization induced by sheafification agrees with the one above). However, since we know that $i^*$ is an equivalence, it implies that this Grothendieck topology must be the trivial one. Notice that we also have an induced adjunction $(i^*,i_*)$ between $\infty$-presheaves. (Here there is no danger of the abuse of notation, since both functors are left exact, so preserves $n$-truncated objects for all $n$, so their restriction to presheaves of sets agree with the ones above). This adjunction is still a left exact localization, and since $Psh_\infty\left(\mathbf{Man}\right)$ is $1$-localic, it must again correspond to a unique Grothendieck topology. This left exact localization factors uniquely as a topological localization (one coming from a Grothendieck topology), followed by a cotopological one (one for which the only monos sent to equivalences are equivalences). Since the $\infty$-category of $\infty$-groupoids is hypercomplete and colimits are computed pointwise in presheaves, $Psh_\infty\left(\mathbf{Man}\right)$ is hypercomplete, so it follows that this localization must be topological. The covering sieves of this topology correspond exactly to those monos $f:S \to y(M)$ such that $i^*(f)$ is an equivalence. However, subobjects of representable objects in $Psh_\infty\left(\mathbf{Man}\right)$ are the same as subobjects in $Psh\left(\mathbf{Man}\right),$ so one sees the resulting class of covering sieves must be the same as for the $1$-categorical case, which we have argued only gave the maximal sieves (the trivial Grothendieck topology). Hence $$i^*:Psh_\infty\left(\mathbf{Man}\right) \to Psh_\infty\left(\mathbf{Open} \right)$$ is an equivalence, and by restriction to $n$-truncated objects,
$$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is as well.