Contact centers process millions of customer conversations daily, requiring Quality Assurance (QA) teams to evaluate agent performance against compliance and service standards, often by answering agent evaluation questionnaires. Traditional manual QA cannot scale to growing volumes, while fully automated evaluation using large language models presents a cost-performance trade-off. High-performing models excel at detecting rare but business-critical Answers of Interest (AoI) but incur prohibitive costs, while smaller fine-tuned models are economical but suffer from poor AoI precision, generating high false positive rates that erode agent trust and waste QA resources. We introduce STREAQ, a two-tier selective routing framework to intelligently route queries between cost-efficient and high-capability models. Based on benchmarking on a proprietary dataset across six large LMs, STREAQ achieves substantial cost reduction while preserving critical performance. Using Nova-Pro, STREAQ reduces daily costs by 48% from 34,162 to17,842 while retaining 88.9% of full-model AoI precision. Our ablation studies reveal that flawed reasoning from smaller models can degrade performance, emphasizing the importance of carefully designing routing systems, making enterprise-scale automated QA both practical and economically viable.