High annotation costs from hiring or crowdsourcing complicate the creation of large, high-quality datasets needed for training reliable text classifiers. Recent research suggests using Large Language Models (LLMs) to automate the annotation process, reducing these costs while maintaining data quality. LLMs have shown promising results in annotating downstream tasks like hate speech detection and political framing. Building on the success in these areas, this study investigates whether LLMs are viable for annotating the complex task of media bias detection and whether a downstream media bias classifier can be trained on such data. We create annolexical, the first large-scale dataset for media bias classification with over 48000 synthetically annotated examples. Our classifier, fine-tuned on this dataset, surpasses all of the annotator LLMs by 5-9 percent in Matthews Correlation Coefficient (MCC) and performs close to or outperforms the model trained on human-labeled data when evaluated on two media bias benchmark datasets (BABE and BASIL). This study demonstrates how our approach significantly reduces the cost of dataset creation in the media bias domain and, by extension, the development of classifiers, while our subsequent behavioral stress-testing reveals some of its current limitations and trade-offs.