In this work, an efficient precoding design scheme is proposed for downlink cell-free distributed massive multiple-input multiple-output (DM-MIMO) filter bank multi-carrier (FBMC) systems with asynchronous reception and highly frequency selectivity. The proposed scheme includes a multiple interpolation structure to eliminate the impact of response difference we recently discovered, which has better performance in highly frequency-selective channels. Besides, we also consider the phase shift in asynchronous reception and introduce a phase compensation in the design process. The phase compensation also benefits from the multiple interpolation structure and better adapts to asynchronous reception. Based on the proposed scheme, we theoretically analyze its ergodic achievable rate performance and derive a closed-form expression. Simulation results show that the derived expression can accurately characterize the rate performance, and FBMC with the proposed scheme outperforms orthogonal frequency-division multiplexing (OFDM) in the asynchronous scenario.