  Vector-tensor theories beyond General Relativity have widely been studied in
the context of ultraviolet completion of gravity, endowing a mass to the
graviton and explaining dark energy phenomena. We here construct rotating black
hole solutions in vector-tensor theories valid after the binary neutron star
merger event GW170817 that placed very stringent bound on the propagation speed
of gravitational waves away from the speed of light. Such valid vector-tensor
theories are constructed by performing a generic conformal transformation to
Einstein-Maxwell theory, and the new rotating black hole solutions are
constructed by applying the same conformal transformation to the Kerr-Newman
solution. These theories fall outside of beyond generalized Proca theories but
are within an extended class of vector-tensor theories that satisfy a
degenerate condition to eliminate instability modes and are thus healthy. We
find that such conformal Kerr-Newman solutions preserve the location of the
singularities, event horizons and ergoregion boundary from Kerr-Newman, as well
as the multipole moments and the Petrov type. On the other hand, the
Hamilton-Jacobi equation is no longer separable, suggesting that the
Carter-like constant does not exist in this solution. The standard Newman-Janis
algorithm also does not work to construct the new solutions. We also compute
the epicyclic frequencies, the location of the innermost stable circular
orbits, and the Schwarzschild precession and apply the latter to the recent
GRAVITY measurement to place bounds on the deviations away from Kerr-Newman for
Sgr A$^*$.
