  We study graviton scattering on $AdS_4\times S^7$, which is dual to the
stress tensor multiplet four-point function in the maximally supersymmetric 3d
$U(N)_1\times U(N)_{-1}$ ABJM theory. We compute 1-loop corrections to this
holographic correlator coming from Witten diagrams with supergravity $R$ and
higher derivative $R^4$ vertices, up to contact term ambiguities, and find that
the flat space limit matches the corresponding terms in the 11d M-theory
S-matrix. We then use supersymmetric localization to show that all the 1-loop
contact terms vanish, as was previously observed for the $AdS_4\times
S^7/\mathbb{Z}_2$ theory dual to $U(N)_2\times U(N)_{-2}$ ABJM. Finally, we use
the recent localization results of Gaiotto and Abajian, as inspired by twisted
M-theory, to compute all the short OPE coefficients in correlators of the
stress tensor multiplet and the next lowest half-BPS operator, which we find
saturate the bootstrap bounds on these mixed correlators for all $N$.
