Abstract: With the rapid development of integrated space–air–ground networks and large-scale constellations, Low Earth Orbit Satellite Networks (LEO-SN) together with Unmanned Aerial Vehicles (UAV) constitute a hierarchical and heterogeneous aerial access/backhaul backbone, realizing three-dimensional “space–air–ground” complementarity and becoming a research hotspot for 6G. Owing to the high packet-loss rate, long latency, and pronounced dynamics of LEO-SN links, the Transmission Control Protocol (TCP) congestion-control mechanisms designed for terrestrial networks cannot maintain stable and efficient bandwidth utilization. Compared with traditional loss-based and delay-based algorithms, probe-based algorithms represented by BBR (Bottleneck Bandwidth and Round-trip propagation time) are better suited to such complex environments; nevertheless, theoretical analysis and experiments in this paper show that, limited by its passive sensing strategy, BBR can suffer a sustained throughput drop during satellite handovers—the so-called “long valley effect”. To address this issue, we propose SHP-BBR, a TCP congestion-control mechanism based on satellite-handover prediction. SHP-BBR leverages historical satellite-handover records and round-trip-time data to build a time-series prediction model that forecasts upcoming handover events. Guided by these predictions, SHP-BBR proactively senses impending link-state changes, obtains accurate network-parameter estimates, and preserves bandwidth utilization. Experimental results demonstrate that SHP-BBR stably eliminates the long valley effect in LEO-SN environments, achieving an average throughput gain of 33%. Moreover, compared with conventional TCP congestion-control algorithms, SHP-BBR delivers efficient, well-balanced throughput and latency under dynamic network conditions.