The rapid proliferation of Large Language Models (LLMs) has created a profound digital divide, effectively excluding indigenous languages of the Global South from the AI revolution. The Tharu language, an Indo-Aryan vernacular spoken by approximately 1.7 million people across the Terai belt of Nepal and India, exemplifies this crisis. Despite a rich oral tradition, Tharu suffers from severe data scarcity and linguistic fragmentation, causing state-of-the-art multilingual models to routinely "hallucinate" or default to dominant high-resource neighbors like Hindi and Nepali due to contamination in pre-training corpora. This paper presents Tharu-LLaMA (3B), a specialized instruction-following model designed to address this exclusion. We introduce TharuChat, a novel dataset constructed via a LLM-to-Human bootstrapping pipeline. We utilized prompt-engineered Gemini models, fed with Rana Tharu grammar and folklore, to synthesize training data. Unlike curated gold-standard corpora, TharuChat reflects the noisy, heterogeneous linguistic reality of the region: it is predominantly anchored in Rana Tharu (~70%) while integrating elements of Dangaura and Kochila dialects. We provide a transparent analysis of the dataset's limitations, including dialectal code-mixing and residual Awadhi/Hindi influence. Through a rigorous empirical ablation study, we demonstrate that despite these imperfections, small-scale synthetic data is highly effective, increasing the dataset volume from 25% to 100% results in a linear reduction in perplexity from 6.42 to 2.88. The resulting model serves as a proof-of-concept for the preservation of under-resourced Himalayan languages via generative AI, achievable on consumer-grade hardware.