Labeling datasets for African languages presents substantial challenges due to the diverse settings in which annotations are collected, resulting in highly variable labeling costs. These costs can vary with task complexity, annotator expertise, and data availability. Yet, most active learning (AL) frameworks assume uniform annotation costs, limiting their applicability in real-world, resource-constrained scenarios. To address this, we introduce KnapsackBALD, a novel cost-aware active learning method that integrates the BatchBALD acquisition strategy with a 0-1 Knapsack optimization objective to select informative and budget-efficient samples. We evaluate KnapsackBALD on the MasakhaNEWS dataset, a multilingual news classification benchmark covering 11 African languages. Our method consistently outperforms seven strong active learning baselines, including BALD, BatchBALD, and stochastic sampling variants such as PowerBALD and Softmax-BALD, across all three cost scenarios. The performance gap widens as annotation cost imbalances become more extreme, demonstrating the robustness of KnapsackBALD under practical constraints. These findings underscore the need for cost-sensitive acquisition in AL pipelines for African language NLP and beyond.