Routing quantum keys over low-earth-orbit (LEO) satellite constellations is harder than classical routing: satellite handovers couple consecutive scheduling decisions, stochastic cloud cover can silently zero a ground link, and finite-key effects eliminate short, low-elevation passes entirely. We present SATLOCK, a handover-aware Quantum Key Distribution (QKD) routing framework that combines (i) a composite channel model incorporating atmospheric loss, pointing jitter, Markov cloud cover, decoy-state estimation, and finite-key correction; (ii) an integer linear program (ILP) giving a provable handover-aware throughput upper bound; and (iii) a decentralized deep Q-network (DQN) baseline for weather-adaptive online routing. We evaluate two contention regimes on a Walker constellation serving intercontinental demands. In low contention (16 satellites, 6 demands), the ILP delivers 1,311 Mbit while the strongest heuristics reach 95--96\% of ILP. In high contention (8 satellites, 12 demands), where handovers become binding, heuristics drop to 89.5\% of ILP. The DQN agent reaches 91.8\% and 84.6\% of ILP in the two regimes; it learns effective per-demand weather policies but is limited in aggregate by the lack of cross-demand coordination.

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