Meta-analyses of observational studies on exposure-disease relationships can serve as the motivation and groundwork for large clinical trials. However, because it is common for each observational study in a meta-analysis to use an imperfect measurement as a surrogate for the true exposure status, observations from each observational study are prone to exposure misclassification. Although misclassification in a single observational study has been well studied, few papers considered it in a meta-analysis. In this paper, we propose a novel Bayesian approach to filling this methodological gap. We simultaneously synthesize two meta-analyses, with one on the association between a misclassified exposure and an outcome, and the other on the association between the misclassified exposure and the true exposure. We extend the current scope of using external validation data by relaxing the ``transportability'' assumption through random effects models. Our model accounts for heterogeneity between studies and can be extended to allow different studies to have different exposure measurements. The proposed model is evaluated through simulations and real-world data.