Non-terrestrial networks (NTNs) increasingly rely on non-geostationary (NGSO) constellations that combine radio frequency (RF) feeder links (FLs) with free space optical (FSO) inter-satellite links (ISLs). Downlink performance in such systems is often constrained by uneven satellite-gateway visibility, data traffic congestion, and rain-induced FL attenuation, leaving the downlink capacity of some satellites underutilized while others become bottlenecks. To prevent such non-uniform load distribution, this paper presents a fairness-driven load balancing strategy that treats the satellite constellation in space as an anycast multi-commodity flow problem. Then, by solving an equivalent linear programming optimization problem, the proposed algorithm dynamically selects the most convenient ground station (GS) to serve each satellite and, when needed, offloads data traffic to adjacent satellites through FSO ISLs. Using a realistic MEO satellite constellation with 1550 nm FSO ISLs and Ka-band feeder links, the method stabilizes the reverse link data service, maintaining the average data rate but notably improving the worst-case throughput. Our proposed algorithm enhances the minimum downlink data rate by more than 25% in the presence of rain and by over 10% under no-rain conditions. These results demonstrate that the use of an ISL-assisted load-balancing scheme mitigates FL bottlenecks and enhances fairness across the satellite constellation, offering a scalable basis for resource allocation in future NTN systems.