Fused filament fabrication (FFF) of continuous fiber-reinforced polymers (FRP) is a promising technology for producing lightweight, strong, and stiff components with fibers aligned along load paths. Despite its benefits, the FFF process for continuous FRPs remains challenging. Achieving three-dimensional load-oriented fiber alignment requires non-planar layers in the printing process, which can be realized through layers of non-uniform thickness. For a layer with a non-uniform thickness, the extrusion rate must be precisely adjusted over a wide range while maintaining process stability. Numerical fluid simulations can enhance the understanding of how key process parameters - such as heating power, extrusion rate, and print head movement speed - affect process stability. However, these simulations are complex due to the intricate melting behavior of thermoplastic polymer within the print head. Accurate modeling must account for high temperature gradients, a phase change from solid to liquid, and a melt with temperature- and shear-rate-dependent viscosity. A solid fiber embedded within the molten thermoplastic introduces additional complexity. Comprehensive analysis is required to understand and model these effects to address process instabilities such as clogging or inconsistent extrusion. In this work, we present a numerical simulation of the molten polymer flow and the associated heating process, aiming to improve and stabilize printhead operation. The model accounts for both temperature and shear-rate dependence of the polymer melt. Simulation results are validated through comparison with data from an existing experimental manufacturing setup. Thus, this study supports the reliable, high-quality FFF of FRP components with variable layer thickness.