Migration has been a core topic in German political debate, from the postwar displacement of millions of expellees to labor migration and recent refugee movements. Studying political speech across such wide-ranging phenomena in depth has traditionally required extensive manual annotation, limiting analysis to small subsets of the data. Large language models (LLMs) offer a potential way to overcome this constraint. Using a theory-driven annotation scheme, we examine how well LLMs annotate subtypes of solidarity and anti-solidarity in German parliamentary debates and whether the resulting labels support valid downstream inference. We first provide a comprehensive evaluation of multiple LLMs, analyzing the effects of model size, prompting strategies, fine-tuning, historical versus contemporary data, and systematic error patterns. We find that the strongest models, especially GPT-5 and gpt-oss-120B, achieve human-level agreement on this task, although their errors remain systematic and bias downstream results. To address this issue, we combine soft-label model outputs with Design-based Supervised Learning (DSL) to reduce bias in long-term trend estimates. Beyond the methodological evaluation, we interpret the resulting annotations from a social-scientific perspective to trace trends in solidarity and anti-solidarity toward migrants in postwar and contemporary Germany. Our approach shows relatively high levels of solidarity in the postwar period, especially in group-based and compassionate forms, and a marked rise in anti-solidarity since 2015, framed through exclusion, undeservingness, and resource burden. We argue that LLMs can support large-scale social-scientific text analysis, but only when their outputs are rigorously validated and statistically corrected.