In this paper, we introduce a novel approach to user-centric association and feedback bit allocation for the downlink of a cell-free massive MIMO (CF-mMIMO) system, operating under limited feedback constraints. In CF-mMIMO systems employing frequency division duplexing, each access point (AP) relies on channel information provided by its associated user equipments (UEs) for beamforming design. Since the uplink control channel is typically shared among UEs, we take account of each AP's total feedback budget, which is distributed among its associated UEs. By employing the Saleh-Valenzuela multi-resolvable path channel model with different average path gains, we first identify necessary feedback information for each UE, along with an appropriate codebook structure. This structure facilitates adaptive quantization of multiple paths based on their dominance. We then formulate a joint optimization problem addressing user-centric UE-AP association and feedback bit allocation. To address this challenge, we analyze the impact of feedback bit allocation and derive our proposed scheme from the solution of an alternative optimization problem aimed at devising long-term policies, explicitly considering the effects of feedback bit allocation. Numerical results show that our proposed scheme effectively enhances the performance of conventional approaches in CF-mMIMO systems.