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Bayesian additive regression trees (BART) provide a class of flexible nonparametric regressions in machine learning. Their rich tree-based structure can generate non-Donsker classes of regression functions. We study posterior inference for the population mean under missing at random. We derive a new Bernstein-von Mises (BvM) theorem showing that even a one-step posterior based on BART and Bayesian-bootstrap reweighting retains a non-negligible bias term in the non-Donsker regime. To remove this term, we introduce RoBART, a posterior correction that adjusts each draw using pilot estimators of the outcome regression and propensity score. We prove a BvM theorem for the corrected posterior and develop a cross-fitted version using fold-specific posteriors. We provide primitive assumptions tailored to BART. We show that the posterior mean coincides exactly with the augmented inverse-probability-weighted (AIPW) estimator and with the corresponding double machine learning (DML) estimator under cross-fitting. RoBART therefore provides a posterior distribution with asymptotically valid uncertainty quantification around this well-established frequentist center. Simulations and an empirical illustration demonstrate competitive finite-sample performance relative to existing methods.