The protection of Intellectual Property (IP) for Large Language Models (LLMs) has become a critical concern as model theft and unauthorized commercialization escalate. While adversarial fingerprinting offers a promising black-box solution for ownership verification, existing methods suffer from significant limitations: they are fragile against downstream model modifications, sensitive to system prompt variations, and easily detectable due to high-perplexity input patterns. In this paper, we propose \textbf{SRAF}, a stealthy and robust adversarial fingerprinting framework. SRAF employs a synergistic joint optimization strategy across homologous model variants and diverse chat templates, forcing the fingerprint to anchor onto the invariant intrinsic comprehension features of the model family. Furthermore, we introduce a Perplexity Hiding technique that embeds adversarial perturbations within Markdown tables, effectively aligning the prompt's statistics with natural language to evade perplexity-based detection. Extensive experiments on the Llama-2 model family demonstrate that SRAF significantly enhances robustness against fine-tuning, alignment, pruning, merging, and input perturbations while maintaining exceptional stealthiness and low false-positive rates, offering a practical and resilient black-box solution for LLM ownership verification.