Automatic Speech Recognition (ASR) can ease clinical documentation in resource-constrained regions, but deployment is hindered by a "Reality Gap" between laboratory performance and noisy, real-world clinical telephony, compounded by strict data residency and compute constraints. We study this gap using Gram Vaani, a telephonic Hindi corpus spanning rural healthcare and agricultural helplines, as the closest publicly available proxy for clinical telephony speech, and show that a robust multilingual model (IndicWav2Vec) degrades from 11.60% WER on clean read Hindi to 41.72% WER on this data. We evaluate a progression of adaptation regimes, from full fine-tuning and offline Low-Rank Adaptation (LoRA) upper bounds to an on-device, stream-based continual adaptation framework in which raw audio never leaves the local device, and characterize the trade-offs between data-driven and parameter-driven stabilization strategies. Our evaluation covers both lexical accuracy (WER and CER) and semantic fidelity (BERTScore) on the target domain, alongside the retention of general-domain knowledge. Multi-domain Experience Replay (ER) yields the primary gains, improving target WER by 18.2% relative and reducing catastrophic forgetting by 54% compared to naive adaptation, with BERTScore reflecting consistent gains in semantic fidelity. Combining replay with Elastic Weight Consolidation based on a stabilized importance estimate (Absolute Fisher) yields the strongest retention at a small cost in plasticity. Finally, a language model spot check empirically verifies that the core mismatch lies at the acoustic level and cannot be resolved by language models alone.

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