  We continue to investigate correspondences between, on the one hand,
scattering amplitudes for massive higher-spin particles and gravitons in
appropriate quantum-to-classical limits, and on the other hand, classical
gravitational interactions of spinning black holes according to general
relativity. We first construct an ansatz for a gravitational Compton amplitude,
at tree level, constrained only by locality, crossing symmetry, unitarity and
consistency with the linearized-Kerr 3-point amplitude, to all orders in the
black hole's spin. We then explore the extent to which a unique classical
Compton amplitude can be identified by comparing with the results of the
classical process of scattering long-wavelength gravitational waves off an
exact Kerr black hole, determined by appropriate solutions of the Teukolsky
equation. Up to fourth order in spin, we find complete agreement with a
previously conjectured exponential form of the tree-level Compton amplitude. At
higher orders, we extract tree-level contributions from the Teukolsky amplitude
by an analytic continuation from a physical ($a/GM<1$) to a particle-like
($a/GM>1$) regime. Up to the sixth order in spin, we identify a unique
\textit{conservative} part of the amplitude which is insensitive both to the
choice of boundary conditions at the black hole horizon and to branch choices
in the analytic continuation. The remainder of the amplitude is determined
modulo an overall sign from a branch choice, with the sign flipping under
exchanging purely ingoing and purely outgoing boundary conditions at the
horizon. Along the way, we make contact with novel applications of massive
spinor-helicity variables pertaining to their relation to EFT operators and
(spinning) partial amplitudes.
