  The ABP method for proving isoperimetric inequalities has been first employed
by Cabr\'e in $\mathbb{R}^n$, then developed by Brendle, notably in the context
of non-compact Riemannian manifolds of non-negative Ricci curvature and
positive asymptotic volume ratio. In this paper, we expand upon their approach
and prove isoperimetric inequalities (sharp in the limit) in the presence of a
small amount of negative curvature. First, we consider smallness of the
negative part $\mathrm{Ric}_-$ of the Ricci curvature in terms of its elliptic
Kato constant. Indeed, the Kato constant turns out to control the
non-negativity of the ($\infty$-)Bakry-\'Emery Ricci-tensor of a suitable
conformal deformation of the manifold, and the ABP method can be implemented in
this setting. Secondly, we show that the smallness of the Kato constant is
implied by a suitable polynomial decay of $\mathrm{Ric}_-$, provided that the
asymptotic volume ratio is positive and either (a) $M$ has one end and
asymptotically non-negative sectional curvature or (b) the relative volume
comparison known as $\textbf{(VC)}$ condition holds. To show this latter fact,
we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of
the Green function valid on the whole manifold.
