In the dynamic construction industry, traditional robotic integration has primarily focused on automating specific tasks, often overlooking the complexity and variability of human aspects in construction workflows. This paper introduces a human-centered approach with a "work companion rover" designed to assist construction workers within their existing practices, aiming to enhance safety and workflow fluency while respecting construction labor's skilled nature. We conduct an in-depth study on deploying a robotic system in carpentry formwork, showcasing a prototype that emphasizes mobility, safety, and comfortable worker-robot collaboration in dynamic environments through a contextual Reinforcement Learning (RL)-driven modular framework. Our research advances robotic applications in construction, advocating for collaborative models where adaptive robots support rather than replace humans, underscoring the potential for an interactive and collaborative human-robot workforce.