This paper deals with uncertainty estimation for random forests, not BART, but I wonder if it might be helpful in an indirect way: Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife
This paper deals with uncertainty estimation for random forests, not BART, but I wonder if it might be helpful in an indirect way: Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife