The main aim of this study was to perform a complex biomechanical analysis for a custom-designed temporomandibular joint (TMJ) prosthesis in combination with a fibular free flap in a pediatric case. Numerical simulations in seven variants of loads were carried out on 3D models obtained based on CT images of a 15-year-old patient in whom it was necessary to reconstruct the temporal-mandibular joints with the use of a fibula autograft. The implant model was designed based on the patient’s geometry. Experimental tests on a manufactured personalized implant were carried out on the MTS Insight testing machine. Two methods of fixing the implant to the bone were analyzed—using three or five bone screws. The greatest stress was located on the top of the head of the prosthesis. The stress on the prosthesis with the five-screw configuration was lower than in the prosthesis with the three-screw configuration. The peak load analysis shows that the samples with the five-screw configuration have a lower deviation (10.88, 0.97, and 32.80%) than the groups with the three-screw configuration (57.89 and 41.10%). However, in the group with the five-screw configuration, the fixation stiffness was relatively lower (a higher value of peak load by displacement of 171.78 and 86.46 N/mm) than in the group with the three-screw configuration (where the peak load by displacement was 52.93, 60.06, and 78.92 N/mm). Based on the experimental and numerical studies performed, it could be stated that the screw configuration is crucial for biomechanical analysis. The results obtained may be an indication for surgeons, especially during planning personalized reconstruction procedures.