Powder bed fusion of metals using a laser beam (PBF-LB/M) is an additive manufacturing process that enables the fabrication of highly complex parts by selectively melting powder material layer by layer. Part-scale overheating remains a critical challenge in PBF-LB/M, particularly for geometries with limited heat dissipation, such as extended overhangs or thin walls. While numerical thermal simulations provide detailed insights into these effects, their high computational cost hinders the widespread application for part-scale analyses and process-aware decision making. This work presents a physics-informed neural network (PINN) framework for the part-scale overheating prediction in PBF-LB/M. By embedding the governing heat conduction equation directly into the learning process, PINNs enable the prediction of temperature fields without the need for prior simulation or experimental datasets. To reduce the computational effort, transfer learning was introduced by exploiting the incremental layer-wise build-up inherent to additive manufacturing. Network states were reused across successive layers, taking advantage of the minor geometric and thermal differences between adjacent layers. This strategy significantly accelerated the convergence while maintaining the prediction accuracy. The framework was evaluated on representative part-scale geometries, demonstrating the ability to identify regions prone to thermal accumulation and overheating. The results show good agreement with the reference numerical simulations. The presented approach highlights the potential of physics-informed machine learning for the efficient part-scale thermal assessment and provides a foundation for the future integration into process and part design workflows.