Training large language models (LLMs) at the network edge faces fundamental challenges arising from device resource constraints, severe data heterogeneity, and heightened privacy risks. To address these, we propose ELSA (Efficient LLM-centric Split Aggregation), a novel framework that systematically integrates split learning (SL) and hierarchical federated learning (HFL) for distributed LLM fine-tuning over resource-constrained edge networks. ELSA introduces three key innovations. First, it employs a task-agnostic, behavior-aware client clustering mechanism that constructs semantic fingerprints using public probe inputs and symmetric KL divergence, further enhanced by prediction-consistency-based trust scoring and latency-aware edge assignment to jointly address data heterogeneity, client unreliability, and communication constraints. Second, it splits the LLM into three parts across clients and edge servers, with the cloud used only for adapter aggregation, enabling an effective balance between on-device computation cost and global convergence stability. Third, it incorporates a lightweight communication scheme based on computational sketches combined with semantic subspace orthogonal perturbation (SS-OP) to reduce communication overhead while mitigating privacy leakage during model exchanges. Experiments across diverse NLP tasks demonstrate that ELSA consistently outperforms state-of-the-art methods in terms of adaptability, convergence behavior, and robustness, establishing a scalable and privacy-aware solution for edge-side LLM fine-tuning under resource constraints.