The magnetic Penrose process (MPP) is one of the most efficient mechanisms for extracting rotational energy from a magnetized Kerr black hole (BH), enabling the acceleration of charged particles up to very-high energies. In this work, we investigate this process in Sagittarius A* (Sgr A), the supermassive black hole (SMBH) at the Galactic center (GC), focusing on its ultrahigh-efficiency regime, in which neutrons undergo beta decay inside the BH ergosphere. As a key novelty, we compute the neutron production spectrum in the accretion flow directly from the underlying nuclear reaction kinematics and follow the neutron trajectories in the Kerr spacetime to determine the population that reaches the ergosphere and undergoes the MPP. From this population, we consistently derive the spectrum of accelerated protons and show that Sgr A can accelerate them up to PeV energies, strengthening its interpretation as a candidate Galactic PeVatron. We further compute the gamma-ray and neutrino emission produced through hadronic interactions of the escaping protons in the Central Molecular Zone. The predicted gamma-ray fluxes exhibit distinctive spectral features that could provide an observational signature of the MPP and may constitute a non-negligible contribution to the very-high-energy emission detected by H.E.S.S. and HAWC. The associated neutrino fluxes remain below the diffuse Galactic component inferred by IceCube, although they may still contribute to the high-energy emission from the GC. Remarkably, the predicted signals lie within the projected sensitivity of SWGO for all the scenarios considered in this work and, for some models, only a factor of a few below the nominal CTAO sensitivity, while future neutrino observatories including KM3NeT/ARCA and IceCube-Gen2 will provide complementary tests of this scenario. Our results establish the MPP as a realistic and observationally testable mechanism for extracting BH rotational energy, while providing a direct connection between horizon-scale physics and high-energy multi-messenger observations that can be naturally extended to other magnetized BHs.