This study explores how large language models (LLMs), specifically GPT-4o, handle semantic ambiguity in low-resource languages, focusing on Hakka (Sfi-Hsien dialect). Unlike previous studies on Taiwanese which emphasize semantic leakage, this paper investigates how LLMs interpret and resolve lexical polysemy, context-dependent meanings, and pragmatically underspecified expressions during Hakka-to-Mandarin AI translation. We introduce the notion of Ambiguity Resolution Trajectories (ART) to trace whether ambiguity is preserved, disambiguated, distorted, or newly generated through back-translation. Our corpus, drawn from the Hakka Language Certification Vocabulary Database, was translated and back-translated using GPT-4o. Through a combined framework of entropy-based stylometrics, embedding divergence, and qualitative content analysis, we categorize ambiguity phenomena and assess AI's pragmatic decision-making. Findings reveal systematic biases in how GPT-4o resolves or simplifies ambiguity, with implications for translation studies, computational pragmatics, and low-resource language equity.