I think you should be able to prove this roughly as follows: first consider the loop space of your construction. For nice simplicial spaces, the loop space can be calculated level-wise (see May's Geometry of Iterated Loop Spaces, for instance), and hence the loop space of your construction is homotopy equivalent to the (realization of the) simplicial space $[n] \mapsto \Omega |\overline{W} Sing_n (G)|$. But this space is level-wise weakly equivalent to $[n] \mapsto Sing_n (G)$, whose realization is homotopy equivalent to $G$. So the loop space of your construction yields $G$, and now you need to apply some form of the statement: $B\Omega X \simeq X$.

Also, note that it's a standard fact (due to Segal's "Classifying spaces and spectral sequences" and/or May's "Classifying spaces and fibrations") that the simplicial bar construction applied directly to your topological group $G$ gives a model for $BG$ (this requires the inclusion of the identity of $G$ to be a cofibration, which is of course true for Lie groups).