The aim of this study was to optimize a set of technological parameters (travel speed, extruder temperature, and extrusion rate) for 3D printing with a PEEK-based composite reinforced with 30 wt.% glass fibers (GFs). For this purpose, both Taguchi and finite element methods (FEM) were utilized. The artificial neural networks (ANNs) were implemented for computer simulation of full-scale experiments. Computed tomography of the additively manufactured (AM) samples showed that the optimal 3D printing parameters were the extruder temperature of 460 °C, the travel speed of 20 mm/min, and the extrusion rate of 4 rpm (the microextruder screw rotation speed). These values correlated well with those obtained by computer simulation using the ANNs. In such cases, the homogeneous micro- and macro-structures were formed with minimal sample distortions and porosity levels within 10 vol.% of both structures. The most likely reason for porosity was the expansion of the molten polymer when it had been squeezed out from the microextruder nozzle. It was concluded that the mechanical properties of such samples can be improved both by changing the 3D printing strategy to ensure the preferential orientation of GFs along the building direction and by reducing porosity via post-printing treatment or ultrasonic compaction.