Conversational AI chatbots have become increasingly common within the customer service industry. Despite improvements in their emotional development, they often lack the authenticity of real customer service interactions or the competence of service providers. By comparing emotion-sensitive and emotion-insensitive LLM-based chatbots across 30 participants, we aim to explore how emotional sensitivity in chatbots influences perceived competence and overall customer satisfaction in service interactions. Additionally, we employ sentiment analysis techniques to analyze and interpret the emotional content of user inputs. We highlight that perceptions of chatbot trustworthiness and competence were higher in the case of the emotion-sensitive chatbot, even if issue resolution rates were not affected. We discuss implications of improved user satisfaction from emotion-sensitive chatbots and potential applications in support services.