We combine two approaches to optimize the iterative decoding of product codes with precoded polar component codes. On one side, we generate bitwise soft messages based on the codebook probability, an approximation of an auxiliary quantity that considers all valid decoding paths of a successive cancellation list (SCL) decoder. On the other side, we scale the soft information during message passing with offline-computed coefficients, which maximize the generalized mutual information (GMI) between the channel input and the outgoing message in each half iteration. Simulation results show significant improvement of the error-correcting performance compared to heuristic scaling and soft information generation based solely on the candidate list of the decoder. Moreover, we present an extrinsic version of the SCL decoder, which we use in a Monte Carlo density evolution analysis to derive decoding thresholds. The computed thresholds accurately predict the performance of the decoder.