Neutrino stacking analysis constrains the paradigm that Gamma-Ray Bursts (GRBs) are the sources of the Ultra-High Energy Cosmic Rays (UHECRs). However, most of the literature has focused on a pure proton composition of UHECRs, which has been disfavored by recent composition measurements of the Pierre Auger Observatory. The injection of nuclei in the sources is here considered by simulating the nuclear cascade in the photon field of a GRB within the (one-zone) internal shock model. We demonstrate that the prompt neutrino flux hardly depends on the injected composition and is potentially able to strongly constrain UHECR models. The cosmogenic neutrino flux expected from cosmic rays escaped from the sources is also computed: nucleons from the disintegration chain in the GRB lead to a significant contribution to this flux, altering the usual expectation related to the heavy composition at injection. A parameter space study within this combined source-propagation model is performed through a fit of the spectrum and composition data of the Pierre Auger Observatory, showing the power of this tool on constraining the features of GRBs or other classes of candidate sources of neutrinos and UHECRs.