We look for a classical double copy structure between gravity and electrodynamics by connecting the descriptions of the scattering of two point masses, and of two point charges, in terms of perturbative (post-Minkowskian or post-Lorentzian) expansions. We do so by recasting available analytical information within the effective-one-body formalism using Kerr-Schild gauges in both cases. Working at the third perturbative level, we find that the usual linear relation (holding in the probe limit) between the dimensionless electric potential, $\tilde{\phi}= \frac{G M}{e_1 e_2} \phi^{\rm el}$, and the Schwarzschildlike gravitational one, $\Phi^{\rm grav}$, is deformed, in the comparable-mass, comparable-charge, case, into a nonlinear relation which becomes universal in the high energy limit: $\Phi^{\rm grav}= 2\tilde{\phi}-5\tilde{\phi}^2 + 18\tilde{\phi}^3$.