Ok, so I have an argument:

First note that the homotopy coherent nerve of $\mathit{Mfd}_W$ is equivalent to the quasicategory obtained by formally inverting $W$ in $N\left(\mathit{Mfd}\right)$- this holds in generality, as shown in a recent paper of Hinich (Proposition 2.2.1 of http://arxiv.org/abs/1311.4128). Let me just be lazy and write $\mathit{Mfd}_W$ also for this quasicategory.

Let $$y:\mathit{Mfd} \hookrightarrow Psh_\infty\left(\mathit{Mfd}\right)$$ be the Yoneda embedding into infinity presheaves. First, note that the localization $y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right)$ at the class of morphisms of the form $y\left(M \times \mathbb{R}\right) \to y(M)$ is canonically equivalent to $Psh_\infty\left(\mathit{Mfd}_W\right).$ To see this observe that the canonical functor $$\mathit{Mfd} \to Psh_\infty\left(\mathit{Mfd}\right) \to y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right)$$ sends $W$ to equivalences, so induces a functor $$\mathit{Mfd}_W \to y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right),$$ which one can show immediately by universal properties exhibits $y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right)$ as the free colimit completion of $\mathit{Mfd}_W$ in the world of quasicategories.

Let $$y_w:\mathit{Mfd}_W \hookrightarrow Psh_\infty\left(\mathit{Mfd}_W\right)\simeq y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right)$$ denote the Yoneda embedding. Uwinding what we've done, we have that $$y_w\left(M\right) \simeq h \circ y\left(M\right)$$ for all manifolds $M.$ Let $h$ denote the localization functor $$h:Psh_\infty\left(\mathit{Mfd}\right) \to y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right).$$ By Lemma 7.5 of http://arxiv.org/abs/1311.3188, we have that $h$ can be computed as:

$$h\left(F\right)\left(M\right) = \operatorname{hocolim} F\left(M \times \Delta^n_{ext}\right)$$

where $\Delta_{ext}:\Delta \to \mathit{Mfd}$ is the "standard cosimplicial manifold," with each $$\Delta^n_{ext} \cong \mathbb{R}^n.$$

Now, $$h\left(y\left(M\right)\right)\left(N\right) = \operatorname{hocolim} Hom\left(N \times \Delta^n_{ext},M\right).$$

This homotopy colimit is a simplicial diagram of sets (regarded as discrete simplicial sets), so the hocolim is simply the starting simplicial set $$Hom\left(N \times \Delta^\star_{ext},M\right).$$ Finally, (and here I guess I am being partly sloppy), this simplicial set is homotopy equivalent to $Sing(M^N),$ where $M^N$ is the compactly generated mapping space. (Incidentally, if anyone has a clean proof of this last claim, let me know).

Notice that $$h\left(y\left(M\right)\right)\left(N\right)\simeq Hom\left(y(N),ih\left(y\left(M\right)\right)\right),$$ where $i$ is the inclusion of $y\left(W\right)^{-1}Psh_\infty\left(\mathit{Mfd}\right)$ into $Psh_\infty\left(\mathit{Mfd}\right),$ and
$$Hom\left(y(N),ih\left(y\left(M\right)\right)\right)\simeq Hom\left(h y(N), h y(M)\right)\simeq Hom\left(y^w(N),y^w(M)\right),$$
and finally $$Hom\left(y^w(N),y^w(M)\right)\simeq Hom_{\mathit{Mfd}_w}\left(N,M\right),$$ since the Yoneda embedding $y^w$ is full and faithful.