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A graph is nonsingular if its adjacency matrix $A\left(G\right)$ is nonsingular. The inverse of a nonsingular graph $G$ is a graph whose adjacency matrix is similar to $A{\left(G\right)}^{-1}$ via a particular type of similarity. Let $\mathscr{H}$ denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in $\mathscr{H}$ which possess unicyclic inverses. We present a characterization of unicyclic graphs in $\mathscr{H}$ which possess bicyclic inverses.