End-to-end design of communication systems using deep autoencoders (AEs) is gaining attention due to its flexibility and excellent performance. Besides single-user transmission, AE-based design is recently explored in multi-user setup, e.g., for designing constellations for non-orthogonal multiple access (NOMA). In this paper, we further advance the design of AE-based downlink NOMA by introducing weighted loss function in the AE training. By changing the weight coefficients, one can flexibly tune the constellation design to balance error probability of different users, without relying on explicit information about their channel quality. Combined with the SICNet decoder, we demonstrate a significant improvement in achievable levels and flexible control of error probability of different users using the proposed weighted AE-based framework.