Context. Very-high-energy gamma-ray observations of the Galactic center (GC) show extended emission that is strongly correlated with the morphology of the central molecular zone (CMZ). The best explanation for that emission is a hadronic interaction between cosmic rays (CRs) and ambient gas, where a CR central and continuous source accelerates protons up to 1 PeV ( PeVatron). However, current models assume very simplistic CR dynamics. Aims. Our goal is to verify if more realistic CR dynamics for the GC environment are consistent with current gamma-ray observations, and whether they could be constrained by upcoming observations with the Cherenkov Telescope Array (CTA). Methods. We generated synthetic gamma-ray maps using a CR transport model with spherical injection, different diffusion regimes (in and out of the CMZ), polar advection, and mono-energetic particles of 1 PeV, and including different CR populations injected from the Arches, Quintuplet, and nuclear clusters of young massive stars, plus supernova Sgr A East. We adopted two different 3D gas distributions consistent with the observed gas column density, either with or without an inner cavity. Results. In order to reproduce the existing observations detected by the High Energy Stereoscopic System (HESS), a ring-like gas distribution, with its mass set by the standard Galactic CO-to-H2 conversion factor, and CR acceleration from all relevant sources are required. For a conversion factor one order of magnitude lower, injection rates that are ten times higher are needed. We show that CTA will be able to differentiate between models with different CR dynamics, proton sources, and CMZ morphologies, owing to its unprecedented sensitivity and angular resolution. Conclusions. More realistic CR dynamics suggest that the CMZ has a large inner cavity and that the GC PeVatron is a composite CR population accelerated by the Arches, Quintuplet, and nuclear star clusters, and Sgr A East.