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Recent work has shown how on-shell three-point amplitudes in gauge theory and gravity, representing the coupling to massive particles, correspond in the classical limit to the curvature spinors of linearised solutions. This connection, made explicit via the KMOC formalism in split metric signature, turns the double copy of scattering amplitudes into the double copy of classical solutions. Here, we extend this framework to the universal massless sector of supergravity, which is the complete double copy of pure gauge theory. Our extension relies on a Riemann-Cartan curvature incorporating the dilaton and the B-field. In this setting, we can determine the most general double copy arising from the product of distinct gauge theory solutions, say a dyon and √Kerr. This gives a double-copy interpretation to gravity solutions of the type Kerr-Taub-NUT-dilaton-axion. We also discuss the extension to heterotic gravity. Finally, we describe how this formalism for the classical double copy relates to others in the literature, namely (i) why it is an on-shell momentum space analogue of the convolutional prescription, and (ii) why a straightforward prescription in position space is possible for certain vacuum solutions.