Online social networks have transformed the ways in which political mobilization messages are disseminated, raising new questions about how peer influence operates at scale. Building on the landmark 61-million-person Facebook experiment \citep{bond201261}, we develop an agent-based simulation framework that integrates real U.S. Census demographic distributions, authentic Twitter network topology, and heterogeneous large language model (LLM) agents to examine the effect of mobilization messages on voter turnout. Each simulated agent is assigned demographic attributes, a personal political stance, and an LLM variant (\texttt{GPT-4.1}, \texttt{GPT-4.1-Mini}, or \texttt{GPT-4.1-Nano}) reflecting its political sophistication. Agents interact over realistic social network structures, receiving personalized feeds and dynamically updating their engagement behaviors and voting intentions. Experimental conditions replicate the informational and social mobilization treatments of the original Facebook study. Across scenarios, the simulator reproduces qualitative patterns observed in field experiments, including stronger mobilization effects under social message treatments and measurable peer spillovers. Our framework provides a controlled, reproducible environment for testing counterfactual designs and sensitivity analyses in political mobilization research, offering a bridge between high-validity field experiments and flexible computational modeling.\footnote{Code and data available at https://github.com/CausalMP/LLM-SocioPol}