We present simulation results of a gradual solar energetic particle (SEP) event detected on 2021 October 9 by multiple spacecraft, including BepiColombo (Bepi) and near-Earth spacecraft such as the Advanced Composition Explorer (ACE). A peculiarity of this event is that the presence of a high-speed stream (HSS) affected the low-energy ion component (≲5 MeV) of the gradual SEP event at both Bepi and ACE, despite the HSS having only a modest solar wind speed increase. Using the EUHFORIA (European Heliospheric FORecasting Information Asset) magnetohydrodynamic model, we replicate the solar wind during the event and the coronal mass ejection (CME) that generated it. We then combine these results with the energetic particle transport model PARADISE (PArticle Radiation Asset Directed at Interplanetary Space Exploration). We find that the structure of the CME-driven shock was affected by the nonuniform solar wind, especially near the HSS, resulting in a shock wave front with strong variations in its properties such as its compression ratio and obliquity. By scaling the emission of energetic particles from the shock to the solar wind compression at the shock, an excellent match between the PARADISE simulation and in situ measurements of ≲5 MeV ions is obtained. Our modeling shows that the intricate intensity variations observed at both ACE and Bepi were influenced by the nonuniform emission of energetic particles from the deformed shock wave and demonstrates the influence of even modest background solar wind structures on the development of SEP events.