High-precision three-dimensional (3D) positioning in dense urban non-line-of-sight (NLOS) environments benefits significantly from cooperation among multiple distributed base stations (BSs). However, forwarding raw CSI from multiple BSs to a central unit (CU) incurs prohibitive fronthaul overhead, which limits scalable cooperative positioning in practice. This paper proposes a learning-based edge-cloud cooperative positioning framework under limited-capacity fronthaul constraints. In the proposed architecture, a neural network is deployed at each BS to compress the locally estimated CSI into a quantized representation subject to a fixed fronthaul payload. The quantized CSI is transmitted to the CU, which performs cooperative 3D positioning by jointly processing the compressed CSI received from multiple BSs. The proposed framework adopts a two-stage training strategy consisting of self-supervised local training at the BSs and end-to-end joint training for positioning at the CU. Simulation results based on a 3.5~GHz 5G NR compliant urban ray-tracing scenario with six BSs and 20~MHz bandwidth show that the proposed method achieves a mean 3D positioning error of 0.48~m and a 90th-percentile error of 0.83~m, while reducing the fronthaul payload to 6.25% of lossless CSI forwarding. The achieved performance is close to that of cooperative positioning with full CSI exchange.