The present study focuses on optimizing GPU offloading techniques within the Peano computational framework for adaptive mesh refinement (AMR) in Euler simulations using the ExaHyPE 2 engine. We first investigate the ExaHyPE 2 finite volumes Rusanov kernels to evolve the system of Euler equations using the CUDA framework. Then, we systematically identify and mitigate performance bottlenecks in the kernels' offloading process, yielding measurable overall improvements in the GPU computational efficiency.