Complex word identification (CWI) is essential in text simplification, yet work on German CWI remains comparatively limited. To address this gap, we investigate the capabilities of three state-of-the-art LLMs and compare them to previously proposed baseline systems. We fine-tune the LLMs in three setups: (i) using the target expression only, (ii) using the target expression together with its sentence-level context, and (iii) using the context and injection of classical machine learning features. Our results show that while pretrained-only LLMs fall short, fine-tuned LLMs set new benchmarks for both binary and probabilistic CWI. In addition, embedding the target in its context sentence improves performance, whereas feature injection has no clearly measurable effect. All models in this paper are trained on the probabilistic CWI task and additionally evaluated on the binary task; thus, we publish a single model that supports both evaluation views We released all accompanying resources (https://github.com/tschomacker/german-cwi-llm) and model checkpoints (https://huggingface.co/collections/tschomacker/german-cwi-llm).