Combining galaxy clustering information from regions of different environmental densities can help break cosmological parameter degeneracies and access non-Gaussian information from the density field that is not readily captured by the standard two-point correlation function (2PCF) analyses. However, modelling these density-dependent statistics down to the non-linear regime has so far remained challenging. We present a simulation-based model that is able to capture the cosmological dependence of the full shape of the density-split clustering (DSC) statistics down to intra-halo scales. Our models are based on neural-network emulators that are trained on high-fidelity mock galaxy catalogues within an extended-Lambda CDM framework, incorporating the effects of redshift-space, Alcock-Paczynski distortions, and models of the halo-galaxy connection. Our models reach sub-per cent level accuracy down to 1 h(-1 )Mpc and are robust against different choices of galaxy-halo connection modelling. When combined with the galaxy 2PCF, DSC can tighten the constraints on omega(cdm), sigma(8), and n(s) by factors of 2.9, 1.9, and 2.1, respectively, compared to a 2PCF-only analysis. DSC additionally puts strong constraints on environment-based assembly bias parameters.