Satellite-aerial-ground integrated network (SAGIN) has been widely envisioned as a promising network architecture for 6G. In the SAGINs, high altitude platform (HAP)-aided relaying satellite systems using hybrid free-space optics (FSO)/radio-frequency (RF) communications have recently attracted research efforts worldwide. Nevertheless, the main drawback of hybrid FSO/RF systems is the restricted bandwidth of the RF connection, especially when the FSO one is blocked by cloud coverage. This paper explores a novel solution for the hybrid FSO/RF HAP-based SAGIN under the impact of weather and atmospheric conditions. Specifically, an additional unmanned aerial vehicle (UAV) is deployed to diverse the FSO link from the HAP-to-ground station to avoid cloud blockage while maintaining a high-speed connection of the FSO link. A mirror array constructed by re-configurable intelligent surface (RIS), an emerging technology, is mounted on the UAV to reflect the signals from the HAP. The channel model of RIS-UAV takes into account both atmospheric turbulence and hovering-induced pointing errors. Furthermore, we present a novel link switching design with a multi-rate adaptation scheme for the proposed network under different weather and turbulence conditions. Numerical results quantitatively confirm the effectiveness of our proposal. Additionally, we provide insightful discussions that can be helpful for the practical system design of RIS-UAV-assisted HAP-based SAGIN using hybrid FSO/RF links. Monte Carlo simulations are also performed to validate the accuracy of theoretical derivations.