Efficient transfer learning methods for large-scale vision–language models (e.g., CLIP) enable strong few-shot transfer, yet existing adaptation methods follow a fixed fine-tuning paradigm that implicitly assumes a uniform importance of the image and text branches, which has not been systematically studied in image classification. Through extensive analysis, we reveal a Branch Bias issue in vision–language image classification: adapting the image encoder does not always improve performance under out-of-distribution settings. Motivated by this observation, we propose A₃B₂, an Adaptive Asymmetric Adapter that alleviates Branch Bias in few-shot learning. A₃B₂ introduces Uncertainty-Aware Adapter Dampening (UAAD), which automatically suppresses image-branch adaptation when prediction uncertainty is high, enabling soft and data-driven control without manual intervention. Architecturally, A₃B₂ adopts a lightweight asymmetric design inspired by mixture-of-experts with Load Balancing Regularization. Extensive experiments demonstrate that A₃B₂ consistently outperforms 11 competitive prompt- and adapter-based baselines.