Accurate coding of tumor diagnoses with ICD-10-GM and ICD-O-3 is essential for structured cancer documentation in Germany. Smaller open-weight LLMs are appealing for privacy-preserving automation but often struggle with coding accuracy in German-language contexts. This study investigates whether instruction-based fine-tuning on public datasets improves the coding accuracy of open-weight LLMs for German tumor diagnosis texts. The evaluation uses coded diagnoses from the local tumor documentation system as test data. In a systematic data quality assessment, the upper limit for ICD-10 coding performance was estimated at 60-79% for exact and 81-94% for partial (three-character codes only) derivation. As training data, over 500,000 question-answer pairs were created based on the ICD-10-GM, ICD-O-3, and OPS catalogues. Eight open-weight models from the Qwen, Llama, and Mistral families (7-70 B parameters) were fine-tuned. ICD-10-GM accuracy rose from 1.4-24% to 41-58%, and partial accuracy from 31-74% to 73-83%. The accuracy of ICD-O-3 topography coding also improved but started and remained considerably lower with an exact accuracy of 22-40% and a partial accuracy of 56-67% after fine-tuning. Malformed code outputs dropped to 0% for all models. Tumor-diagnosis recognition reached 99%. Accuracy correlated positively with model size, but gaps between small and large models narrowed after fine-tuning. The reasoning mode in Qwen3 generally yielded a lower performance than fine-tuning and was over 100 times slower. Our findings highlight the potential of leveraging public catalogues to build instruction datasets that improve LLMs in medical documentation tasks. The complete training dataset and the best-performing checkpoints of the fine-tuned models are available from https://huggingface.co/datasets/stefan-m-lenz/ICDOPS-QA-2024.