Cell-free massive MIMO (CF-mMIMO) systems promise enhanced spectral efficiency and uniform service by deploying numerous distributed access points (APs) that jointly serve users. However, centralized precoding, which offers optimal performance by leveraging global channel state information (CSI), imposes significant fronthaul overhead due to the extensive CSI exchange between APs and the central processing unit (CPU). In contrast, distributed precoding reduces fronthaul demands, but undergoes performance degradation due to the lack of coordination between APs. This paper introduces a novel framework that strikes a balance between these two scenarios by employing channel charting (CC) as a lightweight CSI compression mechanism at each AP. CC maps high-dimensional CSI into a low-dimensional pseudo-location space, capturing essential channel characteristics. These pseudo-locations are transmitted to the CPU with minimal overhead, where advanced machine learning models are employed to infer effective downlink precoding vectors, thereby approximating centralized performance while reducing fronthaul signaling. Extensive simulations on realistic synthetic datasets demonstrate that our approach achieves very promising results. This work highlights the potential of integrating CC and machine learning to enhance scalability and efficiency in CF-mMIMO networks.