Existing LLM routing frameworks treat queries as independent events, neglecting the sequential nature of real-world user sessions constrained by global computational budgets. This mismatch inevitably leads to budget bankruptcy: myopic routing policies exhaust resources on early interactions, forcing subsequent and often more complex queries onto inadequate models. We introduce SeqRoute, a framework that formulates multi-turn routing as a finite-horizon Markov Decision Process and solves it via offline reinforcement learning. By incorporating the remaining budget into the state space and training with Conservative Q-Learning (CQL), SeqRoute learns delayed gratification to strategically preserve resources for high-stakes turns later in the session. To overcome data starvation, we propose Hindsight Budget Relabeling (HBR). This technique retrospectively simulates historical trajectories under diverse hypothetical budgets, expanding 10,000 raw sessions into 2.38 million transitions enriched with critical bankruptcy signals. At deployment, a dynamic $λ$-sweep mechanism enables zero-shot navigation of the cost-quality Pareto frontier without retraining. Extensive evaluations demonstrate that SeqRoute reduces operational costs by 6.0-73.5% while maintaining or improving quality, and suppresses bankruptcy rates to under 1%, strictly dominating behavior cloning, budget-aware heuristics, and static baselines across the entire Pareto frontier.