Recent advances in quantum computing and machine learning have given rise to quantum machine learning (QML), with growing interest in learning from sequential data. Quantum recurrent models like QLSTM are promising for time-series prediction, NLP, and reinforcement learning. However, designing effective variational quantum circuits (VQCs) remains challenging and often task-specific. To address this, we propose DiffQAS-QLSTM, an end-to-end differentiable framework that optimizes both VQC parameters and architecture selection during training. Our results show that DiffQAS-QLSTM consistently outperforms handcrafted baselines, achieving lower loss across diverse test settings. This approach opens the door to scalable and adaptive quantum sequence learning.