Monocular height estimation provides an efficient and cost-effective solution for three-dimensional perception in remote sensing. However, training deep neural networks for this task demands abundant annotated data, while high-quality labels are scarce and typically available only in developed regions, which limits model generalization and constrains their applicability at large scales. This work addresses the problem by leveraging imperfect labels from out-of-domain regions to train pixel-wise height estimation networks, which may be incomplete, inexact, or inaccurate compared to high-quality annotations. We introduce an ensemble-based pipeline compatible with any monocular height estimation network, featuring architecture and loss functions specifically designed to leverage information in noisy labels through weak supervision, utilizing balanced soft losses and ordinal constraints. Experiments on two datasets -- DFC23 (0.5--1 m) and GBH (3 m) -- show that our method achieves more consistent cross-domain performance, reducing average RMSE by up to 22.94% on DFC23 and 18.62% on GBH compared with baselines. Ablation studies confirm the contribution of each design component.