Understanding whether and how treatment effects vary across individuals 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, under-coverage, inflated type I error, or low power. While statistical methods for handling missing data have been proposed and compared for individually randomized trials with missing effect modifiers, 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 paper, the performance of various missing data methods are neutrally compared in a simulation study of cluster-randomized trials with missing effect modifier data, and a Bayesian multilevel multiple imputation approach is proposed and evaluated. Thereafter, we impose controlled missing data scenarios to potential effect modifiers from the Work, Family, and Health Study to illustrate the proposed missing data method.