We study a Wyner-type downlink network, where any two adjacent base stations (BSs) can communicate with each other and cooperate on transmission of discrete data symbols to their intended mobile stations (MSs) situated along a circular array. With focus on the binary case, we take the symbol detection point of view and formulate a novel non-linear precoding optimization problem by imposing minimum Symbol Error Probabilities as Quality-of-Service constraints at MSs. We then cast the problem into a factor graph and propose a belief propagation (BP) framework to solve the problem in a distributed manner based on only local communication between adjacent BSs. Numerical results confirm the convergence of the distributed algorithm and demonstrate the transmit power saving from the proposed precoder over two state-of-the-art linear precoders. To the best of our knowledge, this is the first work that leverages the BP method for non-linear precoding.