Multimodal respiratory sound classification offers promise for early pulmonary disease detection by integrating bioacoustic signals with patient metadata. Nevertheless, current approaches remain vulnerable to spurious correlations from attributes such as age, sex, or acquisition device, which hinder their generalization, especially under distribution shifts across clinical sites. To this end, we propose a counterfactual adversarial debiasing framework. First, we employ a causal graph-based counterfactual debiasing methodology to suppress non-causal dependencies from patient metadata. Second, we introduce adversarial debiasing to learn metadata-insensitive representations and reduce metadata-specific biases. Third, we design counterfactual metadata augmentation to mitigate spurious correlations further and strengthen metadata-invariant representations. By doing so, our method consistently outperforms strong baselines in evaluations under both in-distribution and distribution shifts. Code is available at https://github.com/RSC-Toolkit/BTS-CARD.