Manufacturing firms increasingly face challenges in inventory allocation due to rising logistics costs, regional demand heterogeneity, and demand fluctuations. To address these issues, this study proposes a demand prediction-based multi-product lateral transshipment location–allocation–inventory (LAI) model for a two-echelon supply chain under a vendor-managed inventory environment. An integrated framework combining an improved grey demand prediction model with the success history-based adaptive differential evolution algorithm (IGDP-SHADE) is developed. The IGDP model is employed to forecast product demand under small-sample and highly volatile conditions, while the SHADE algorithm optimizes facility location, inventory allocation, replenishment, and lateral transshipment decisions with the objective of minimizing total system costs. A case study based on the excavator supply chain of Sany Heavy Industry is conducted to evaluate the proposed approach. The results demonstrate that the IGDP model achieves a mean absolute percentage error (MAPE) of 13.86% and a root mean squared logarithmic error (RMSLE) of 0.1806, outperforming benchmark models including ARIMA and Holt–Winters in forecasting seasonal small-sample demand data. Compared with Gurobi, the proposed IGDP-SHADE algorithm obtains the same optimal solution while reducing computational time by 17.47%. Sensitivity results indicate that introducing the lateral transshipment mechanism reduces total system costs by 2.97% compared with the scenario without transshipment. Furthermore, the optimal number of centralized storage points is identified as six, enabling an effective balance between transportation, inventory holding, and transshipment costs.