Large Language Models (LLMs) demonstrate significant persuasive capabilities in one-on-one interactions, but their influence within social networks, where interconnected users and complex opinion dynamics pose unique challenges, remains underexplored. This paper addresses the research question: \emph{Can LLMs generate meaningful content that maximizes user engagement on social networks?} To answer this, we propose a pipeline using reinforcement learning with simulated feedback, where the network's response to LLM-generated content (i.e., the reward) is simulated through a formal engagement model. This approach bypasses the temporal cost and complexity of live experiments, enabling an efficient feedback loop between the LLM and the network under study. It also allows to control over endogenous factors such as the LLM's position within the social network and the distribution of opinions on a given topic. Our approach is adaptive to the opinion distribution of the underlying network and agnostic to the specifics of the engagement model, which is embedded as a plug-and-play component. Such flexibility makes it suitable for more complex engagement tasks and interventions in computational social science. Using our framework, we analyze the performance of LLMs in generating social engagement under different conditions, showcasing their full potential in this task. The experimental code is publicly available at https://github.com/mminici/Engagement-Driven-Content-Generation.