Live entertainment is moving towards a greater participatory culture, with dynamic narratives told through audience interaction. Robot characters offer a unique opportunity to mitigate the challenges of creating personalized entertainment at scale. However, robots often cannot react to audience responses, limiting opportunities for audience participation. In this work, we explore methods to increase user agency in live entertainment experiences with robot characters to improve user engagement and enjoyment. In a between-subjects study (N=60), we create an immersive story where users role-play as detectives with two distinct robot characters. Users either (1) have greater involvement and self-identification in the story by talking with the robots in-character (narrative condition), (2) have a more active role in solving puzzles (gameplay condition), or (3) follow along without being prompted by the robots for input (control condition). Our results show that increasing user agency in a role-playing experience, in either its narrative or its gameplay, improves users' flow state, sense of autonomy and competence, verbal engagement, and perceptions of the robot characters' engagement. Increasing narrative agency also led to longer unprompted reactions from participants, while gameplay agency improved feelings of immersion and relatedness with the robots. These findings suggest that creating either narrative or gameplay agency can improve user engagement, which can extend to broader robot interactions where gameplay elements and role-playing in stories can be incorporated.