  We analyze correlation functions of $SU(k) \times SU(2)_F$ flavor currents in
a family of three-dimensional ${\cal N}=4$ superconformal field theories,
combining analytic bootstrap methods with input from supersymmetric
localization. Via holographic duality, we extract gluon and graviton scattering
amplitudes of M-theory on ${\rm AdS}_4\times S^7/\mathbb{Z}_k$ which contains a
$\mathbb{C}^2/\mathbb{Z}_{k}$ orbifold singularity. From these results, we
derive aspects of the effective description of M-theory on the orbifold
singularity beyond its leading low energy limit. We also determine a threshold
correction to the holographic correlator from the combined contribution of
two-loop gluon and tree-level bulk graviton exchange.
