The authors propose RQ-TTSA, a distribution-aware framework designed to address instability in bilevel optimization caused by heavy-tailed stochastic noise. Unlike existing variance-reduction techniques that rely on myopic magnitude checks, this method uses historical gradient buffers to estimate rolling quantiles for adaptive Huber-style clipping. This approach preserves local optimization geometry while strictly bounding effective variance under nonconvex-strongly convex assumptions with infinite-variance noise. Theoretical analysis derives a convergence rate of O(T^(-(p-1)/(3p-2))) that recovers optimal dependence on the heavy-tailed parameter p. Empirical evaluations across six diverse tasks, including vision benchmarks and offline reinforcement learning, show consistent outperformance over state-of-the-art baselines. RQ-TTSA eliminates divergence spikes and ensures stable convergence with negligible computational overhead of approximately 2.7 percent.