The evolution of the Industrial Internet of Things (IIoT) and Edge Computing enables resource-constrained mobile robots to offload the computationally intensive localization algorithms. Naturally, utilizing the remote resources of an edge server to offload these tasks, encounters the challenge of a joint co-design in communication, control, estimation and computing infrastructure. We introdce a set-based estimation offloading framework, for the specific case of the navigation of a unicycle robot towards a target position. The robot is subject to modeling and measurement uncertainties, and the estimation set is calculated using overapproximation techniques that alleviate additional computations. A switching set-based control mechanism provides accurate navigation and triggers more precise estimation algorithms when needed. To guarantee the convergence of the system and optimize the utilization of remote resources, a utility-based offloading mechanism is designed, which takes into account both the dynamic network conditions and the available computing resources at the network edge. The performance of the proposed framework is demonstrated through simulations and comparison with alternative offloading schemes.