Recent studies have shown that few-shot learning enables large language models (LLMs) to generate training data for supervised models at a low cost. However, for complex tasks, the quality of LLM-generated data often falls short compared to human-labeled data. This presents a critical challenge: how should one balance the trade-off between the higher quality but more expensive human-annotated data and the lower quality yet significantly cheaper LLM-generated data? In this paper, we tackle this question for a demanding task: conversational semantic frame analysis (SFA). To address this, we propose a novel method for synthesizing training data tailored to this complex task. Through experiments conducted across a wide range of budget levels, we find that smaller budgets favor a higher reliance on LLM-generated data to achieve optimal cost-efficiency.