Bin-picking of metal objects using low-cost RGB-D cameras often suffers from sparse depth information and reflective surface textures, leading to errors and the need for manual labeling. To reduce human intervention, we propose a two-stage framework consisting of a metric learning stage and a self-training stage. Specifically, to automatically process data captured by a low-cost camera (LC), we introduce a Multi-object Pose Reasoning (MoPR) algorithm that optimizes pose hypotheses under depth, collision, and boundary constraints. To further refine pose candidates, we adopt a Symmetry-aware Lie-group based Bayesian Gaussian Mixture Model (SaL-BGMM), integrated with the Expectation-Maximization (EM) algorithm, for symmetry-aware filtering. Additionally, we propose a Weighted Ranking Information Noise Contrastive Estimation (WR-InfoNCE) loss to enable the LC to learn a perceptual metric from reconstructed data, supporting self-training on untrained or even unseen objects. Experimental results show that our approach outperforms several state-of-the-art methods on both the ROBI dataset and our newly introduced Self-ROBI dataset.