Despite significant advancements in terrestrial networks, inherent limitations persist in providing reliable coverage to remote areas and maintaining resilience during natural disasters. Multi-tier networks with low Earth orbit (LEO) satellites and high-altitude platforms (HAPs) offer promising solutions, but face challenges from high mobility and dynamic channel conditions that cause unstable connections and frequent handovers. In this paper, we design a three-tier network architecture that integrates LEO satellites, HAPs, and ground terminals with hybrid free-space optical (FSO) and radio frequency (RF) links to maximize coverage while maintaining connectivity reliability. This hybrid approach leverages the high bandwidth of FSO for satellite-to-HAP links and the weather resilience of RF for HAP-to-ground links. We formulate a joint optimization problem to simultaneously balance downlink transmission rate and handover frequency by optimizing network configuration and satellite handover decisions. The problem is highly dynamic and non-convex with time-coupled constraints. To address these challenges, we propose a novel large language model (LLM)-guided truncated quantile critics algorithm with dynamic action masking (LTQC-DAM) that utilizes dynamic action masking to eliminate unnecessary exploration and employs LLMs to adaptively tune hyperparameters. Simulation results demonstrate that the proposed LTQC-DAM algorithm outperforms baseline algorithms in terms of convergence, downlink transmission rate, and handover frequency. We also reveal that compared to other state-of-the-art LLMs, DeepSeek delivers the best performance through gradual, contextually-aware parameter adjustments.