We study the rainbow matching (RM) problem: given an edge-colored graph, find a maximum matching with at most one edge of each color. Rainbow matchings correspond to stable sets in the \emph{augmented} graph $H$ obtained from the line graph by completing each color class into a clique. For a hereditary graph class $\mathcal{X}$, we introduce the parameter $κ_{\mathcal{X}}$ to be the minimum number of colors whose deletion places the \emph{residual} augmented graph in $\mathcal{X}$. We show that this parameter has two complementary flavors. From a polyhedral side, if $\mathcal{X}$ is uniformly rank-$r$ exact, then deleting $k$ colors to obtain a residual augmented graph in $\mathcal{X}$ implies exactness of the Lasserre hierarchy at level $k+r$. This yields, in particular, exactness at level $k+1$ for deletion to perfect, and exactness at level $k+r$ for deletion to $h$-perfect residual graphs of bounded odd-hole rank $r$. Our second result is structural. We show that the right object in this case is the \emph{color-intersection} graph $Γ$ that impacts the topology of the conflict graph $H$ as follows: articulation colors in $Γ$ induce clique-sum decompositions in $H$, so residual obstructions for clique-sum-local hereditary classes $\mathcal{X}$ are embedded in individual blocks. Thus we can test membership of the residual graph in these target classes in a blockwise manner. As a consequence, we give an exact dynamic programming algorithm for computing the deletion parameter when $Γ$ has blocks of bounded size. Finally, once such a deletion set is given, RM can be solved by branching over the deleted color classes and solving residual instances. We also show that computing this parameter is \textbf{NP}-hard already in the chordal targets but it is FPT for classes $\mathcal{X}$ characterized by a set of forbidden induced subgraphs of bounded size.