Unmanned Aerial Vehicle (UAV) cluster services are crucial for promoting the low-altitude economy by enabling scalable, flexible, and adaptive aerial networks. To meet diverse service demands, clusters must dynamically incorporate a New UAVs (NUAVs) or an Existing UAV (EUAV). However, achieving sustained service reliability remains challenging due to the need for efficient and scalable NUAV authentication, privacy-preserving cross-cluster authentication for EUAVs, and robust protection of the cluster session key, including both forward and backward secrecy. To address these challenges, we propose a Lightweight and Privacy-Preserving Cluster Authentication and Session Key Update (LP2-CASKU) scheme tailored for dynamic UAV clusters in low-altitude economy networks. LP2-CASKU integrates an efficient batch authentication mechanism that simultaneously authenticates multiple NUAVs with minimal communication overhead. It further introduces a lightweight cross-cluster authentication mechanism that ensures EUAV anonymity and unlinkability. Additionally, a secure session key update mechanism is incorporated to maintain key confidentiality over time, thereby preserving both forward and backward secrecy. We provide a comprehensive security analysis and evaluate LP2-CASKU performance through both theoretical analysis and OMNeT++ simulations. Experimental results demonstrate that, compared to the baseline, LP2-CASKU achieves a latency reduction of 82.8%-90.8% by across different UAV swarm configurations and network bitrates, demonstrating strong adaptability to dynamic communication environments. Besides, under varying UAV swarm configurations, LP2-CASKU reduces the energy consumption by approximately 37.6-72.6%, while effectively supporting privacy-preserving authentication in highly dynamic UAV cluster environments.