  We consider heat operators on a bounded domain $\Omega \subseteq
\mathbb{R}^n$, with a critically singular potential diverging as the inverse
square of the distance to $\partial \Omega$. While null boundary
controllability for such operators was recently proved in all dimensions in
arXiv:2112.04457, it crucially assumed (i) $\Omega$ was convex, (ii) the
control must be prescribed along all of $\partial \Omega$, and (iii) the
strength of the singular potential must be restricted to a particular subrange.
In this article, we prove instead a definitive approximate boundary control
result for these operators, in that we (i) do not assume convexity of $\Omega$,
(ii) allow for the control to be localized near any $x_0 \in \partial \Omega$,
and (iii) treat the full range of strength parameters for the singular
potential. Morever, we lower the regularity required for $\partial \Omega$ and
the lower-order coefficients. The key novelty is a local Carleman estimate near
$x_0$, with a carefully chosen weight that takes into account both the
appropriate boundary conditions and the local geometry of $\partial \Omega$.
