Thematic Analysis (TA) is a fundamental method in healthcare research for analyzing transcript data, but it is resource-intensive and difficult to scale for large, complex datasets. This study investigates the potential of large language models (LLMs) to augment the inductive TA process in high-stakes healthcare settings. Focusing on interview transcripts from parents of children with Anomalous Aortic Origin of a Coronary Artery (AAOCA), a rare congenital heart disease, we propose an LLM-Enhanced Thematic Analysis (LLM-TA) pipeline. Our pipeline integrates an affordable state-of-the-art LLM (GPT-4o mini), LangChain, and prompt engineering with chunking techniques to analyze nine detailed transcripts following the inductive TA framework. We evaluate the LLM-generated themes against human-generated results using thematic similarity metrics, LLM-assisted assessments, and expert reviews. Results demonstrate that our pipeline outperforms existing LLM-assisted TA methods significantly. While the pipeline alone has not yet reached human-level quality in inductive TA, it shows great potential to improve scalability, efficiency, and accuracy while reducing analyst workload when working collaboratively with domain experts. We provide practical recommendations for incorporating LLMs into high-stakes TA workflows and emphasize the importance of close collaboration with domain experts to address challenges related to real-world applicability and dataset complexity. https://github.com/jiaweixu98/LLM-TA