This paper optimizes the fronthaul bit allocation in massive multi-user multiple-input multiple-output (MU-MIMO) systems operating with limited-capacity fronthaul links. We consider an advanced antenna system (AAS) controlled by a centralized baseband unit (BBU). In the AAS, multiple antenna elements together with their radio units are integrated into a single unit. In this setup, a key challenge is allocating fronthaul bits between uplink channel state information (CSI) quantization and downlink precoding matrix quantization. We formulate the problem of maximizing the sum spectral efficiency (SE) for a given fronthaul capacity. We develop an SE expression for this scenario based on the hardening bound. We compute the expression in closed form for maximum ratio transmission, which reveals the relative impact of the two types of quantization distortion. We then formulate a bit split optimization problem and propose an algorithm that exactly solves it. Numerical results demonstrate how the relative importance of assigning bits to CSI and precoding varies depending on the signal-to-noise ratio.