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An outstanding problem in heliospheric physics is understanding the acceleration of solar energetic particles (SEPs) in coronal mass ejections (CMEs) and flares. A fundamental question is whether the acceleration occurs in interplanetary space or near the Sun. Recent work has shown that CME-driven shocks may produce SEPs while still below five solar radii. In this work we explore SEP acceleration during the onset of CMEs and shocks even lower in the corona, using realistic suprathermal spectra, for a selection of events. We have calculated quiet-time, pre-event suprathermal particle spectra from 1 au observations, and scaled them back to the low corona to serve as seed spectra. For each event, AIA observations and the CASHeW framework were used to model the compressive/shock wave kinematics and its interaction with the corona. The proton acceleration was then modeled using an analytic diffusive shock acceleration model as the shock waves propagate between ∼1.05 and ∼1.3 solar radii. We demonstrate the capability of low coronal shock-related extreme ultraviolet waves to accelerate protons to multi-MeV energies in a matter of minutes, in the very early stages of the associated solar eruptions. We find that strong proton energization occurs for high values of the density jump, Alfvén Mach number, and shock speed. In future work the results of this early-stage shock acceleration will be used to model the continued acceleration higher in the corona. [ABSTRACT FROM AUTHOR]