The integration of electromagnetic metasurfaces into wireless communications enables intelligent control of the propagation environment. Recently, flexible intelligent metasurfaces (FIMs) have evolved beyond conventional reconfigurable intelligent surfaces (RISs), enabling three-dimensional surface deformation for adaptive wave manipulation. However, most existing FIM-aided system designs assume perfect instantaneous channel state information (CSI), which is impractical in large-scale networks due to the high training overhead and complicated channel estimation. To overcome this limitation, we propose a robust statistical-CSI-based optimization framework for downlink multiple-input single-output (MISO) systems with FIM-assisted transmitters. A block coordinate ascent (BCA)-based iterative algorithm is developed to jointly optimize power allocation and FIM morphing, maximizing the average achievable sum rate. Simulation results show that the proposed statistical-CSI-driven FIM design significantly outperforms conventional rigid antenna arrays (RAAs), validating its effectiveness and practicality.