Accreting supermassive binary black holes (SMBBHs) are potential targets for multi-messenger astronomy as they emit gravitational waves (GW) while their environment emits electromagnetic (EM) waves. In order to get the most out of a joint GW-EM detection we first need to obtain theoretically-predicted EM signals unambiguously linked to BBHs. In that respect, this is the first of a series of papers dedicated to accreting pre-merger BBHs and their associated EM observables. Here, we extend our Numerical Observatory of Violent Accreting systems, e-NOVAs, to any spacetime. Unlike previous studies, almost exclusively focused on the inner regions, we investigated the impact of the BBH on its outer circumbinary disc, located in the radiation (or wave) zone, after implementing an approximate analytical spacetime of spinning, inspiralling BBHs in e-NOVAs. We follow the formation of a weak spiral structure in disc density arising from the retardation effects in the radiation zone metric. Simulation data are then post-processed with a general-relativistic ray-tracing code incorporating the same BBH spacetime, assuming SMBBH sources. The density spiral creates a small (<1%) but unambiguous modulation of the lightcurve at the semi-orbital period. This signal, although weak, is fundamentally different from that of an axisymmetric disc around a single BH providing a lower limit on the impact of a BBH on its outer disc. This potential difference being found, we study how binary parameters impact this modulation in order to find the optimal case which is a high source inclination of any binary mass ratio (from 0.1 to 1).