Many retail distribution networks use a hierarchical structure in which regional distribution centers replenish local distribution centers that are closer to customers. In such networks, products are often shipped in large quantities, as case packs, cases or pallets, to the regional distribution center, whereas local distribution centers may require quantities at the individual-unit level. Case breaking, which disaggregates case packs into individual units at the regional distribution center, enables more flexible allocation across local distribution centers. However, limited processing capacity may prevent all cases from being broken upstream. Planners must therefore decide which cases to break and how to allocate both individual units and intact cases. Despite its practical importance, there is scant literature on explicitly incorporating case-breaking capacity into inventory allocation models. Motivated by this gap, we study an operational case-pack allocation problem from a regional to local distribution centers with case-breaking capacity constraints. Given available stock at a regional distribution center, we optimize a concave surrogate objective function that captures product urgency, current inventory status, and expected demand at local distribution centers. We formulate the problem as an integer program and propose a column generation algorithm with efficient pricing heuristics. We develop an efficient iterative greedy algorithm and a dynamic programming-based heuristic that gradually considers larger action spaces over iterations. Computational experiments on industrial-scale instances show that the method finds numerically optimal solutions, with an average relative optimality gap under 0.0001 in roughly 100 minutes, outperforming a commercial solver in both solution quality and runtime. We further analyze how the allocation plan changes under different capacity constraints and individual-unit handling requirements.