Quantum computing devices are being installed alongside supercomputing clusters to serve as hardware accelerators. This new type of architecture will require an integrated hybrid software stack. With this goal in mind, we have developed a fully integrated hybrid quantum-classical method in the hope of improving Register Allocation in the classical LLVM compiler. We propose a hybrid variational optimization algorithm for the PBQP formulation of Register Allocation. We implemented this algorithm in C++ inside LLVM using the NVIDIA CUDA-Q framework. The performance of the method is evaluated using NVIDIA CUDA-Q noiseless emulators and shows promising results while still needing further optimizations. Our work constitutes a demonstration of an end-to-end tight integration of a quantum subroutine inside an existing classical codebase of interest with potentially interesting performance in fault-tolerant hardware.