  In recent work (Maierhofer & Huybrechs, 2022, Adv. Comput. Math.), the
authors showed that least-squares oversampling can improve the convergence
properties of collocation methods for boundary integral equations involving
operators of certain pseudo-differential form. The underlying principle is that
the discrete method approximates a Bubnov$-$Galerkin method in a suitable
sense. In the present work, we extend this analysis to the case when the
integral operator is perturbed by a compact operator $\mathcal{K}$ which is
continuous as a map on Sobolev spaces on the boundary,
$\mathcal{K}:H^{p}\rightarrow H^{q}$ for all $p,q\in\mathbb{R}$.
  This study is complicated by the fact that both the test and trial functions
in the discrete Bubnov-Galerkin orthogonality conditions are modified over the
unperturbed setting. Our analysis guarantees that previous results concerning
optimal convergence rates and sufficient rates of oversampling are preserved in
the more general case. Indeed, for the first time, this analysis provides a
complete explanation of the advantages of least-squares oversampled collocation
for boundary integral formulations of the Laplace equation on arbitrary smooth
Jordan curves in 2D. Our theoretical results are shown to be in very good
agreement with numerical experiments.
