In recent years, the integration of artificial intelligence (AI) into Learning Management Systems (LMS) has expanded, though many tools lack alignment with learning environment competencies, causing cognitive overload and diminishing learner motivation. This paper explores integrating an AI-based formative feedback system within an LMS, designed to enhance learner competencies and align with curriculum and learning objectives. Utilizing a self-trained large language model, the system processes predefined teaching materials to deliver efficient formative feedback, minimizing cognitive overload. Our study, involving 450 students from the University of Leipzig, assesses the impact of AI-generated feedback during lectures on learning performance, validated against specified learning objectives and competencies. Results emphasize the importance of incorporating sound pedagogical practices in developing AI tools for adaptive learning environments, demonstrating that AI-driven formative feedback can significantly enhance the efficacy of these environments and achieve educational goals.