We develop a radiation pressure-balanced model for the interstellar medium of high-redshift galaxies that describes many facets of galaxy formation at z ≳ 6, including star formation rates and distributions and gas accretion on to central black holes. We first show that the vertical gravitational force in the disc of such a model is dominated by the disc self-gravity supported by the radiation pressure of ionizing starlight on gas. Constraining our model to reproduce the UV luminosity function of Lyman-break galaxies (LBGs), we limit the available parameter space to wind mass-loading factors one to four times the canonical value for momentum-driven winds. We then focus our study by exploring the effects of different angular momentum transport mechanisms in the galactic disc and find that accretion driven by gravitational torques, such as from linear spiral waves or non-linear orbit crossings, can build up black hole masses by z = 6 consistent with the canonical M–σ relation with a duty cycle of unity, while accretion mediated by a local viscosity such as in an α-disc results in negligible black hole (BH) accretion. Both gravitational torque models produce X-ray emission from active galactic nuclei (AGN) in high-redshift LBGs in excess of the estimated contribution from high-mass X-ray binaries. Using a recent analysis of deep Chandra observations by Cowie et al., we can already begin to rule out the most extreme regions of our parameter space: the inflow velocity of gas through the disc must either be less than one per cent of the disc circular velocity or the X-ray luminosity of the AGN must be substantially obscured. Moderately deeper future observations or larger sample sizes will be able to probe the more reasonable range of angular momentum transport models and obscuring geometries.