A recently proposed FENSHSES method from [1] seems to be the state-of-the-art one to conduct r-neighbor search in Hamming space on Elasticsearch.
[1] Mu, C, Zhao, J., Yang, G., Yang, B. and Yan, Z., 2019, October. Fast and Exact Nearest Neighbor Search in Hamming Space on Full-Text Search Engines. In International Conference on Similarity Search and Applications (pp. 49-56). Springer, Cham.