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Understanding whether and how treatment effects vary across subgroups is crucial to inform clinical practice and recommendations. Accordingly, the assessment of heterogeneous treatment effects (HTE) based on pre-specified potential effect modifiers has become a common goal in modern randomized trials. However, when one or more potential effect modifiers are missing, complete-case analysis may lead to bias and under-coverage. While statistical methods for handling missing data have been proposed and compared for individually randomized trials with missing effect modifier data, few guidelines exist for the cluster-randomized setting, where intracluster correlations in the effect modifiers, outcomes, or even missingness mechanisms may introduce further threats to accurate assessment of HTE. In this article, the performance of several missing data methods are compared through a simulation study of cluster-randomized trials with continuous outcome and missing binary effect modifier data, and further illustrated using real data from the Work, Family, and Health Study. Our results suggest that multilevel multiple imputation (MMI) and Bayesian MMI have better performance than other available methods, and that Bayesian MMI has lower bias and closer to nominal coverage than standard MMI when there are model specification or compatibility issues.