We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equationsand also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces.Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping Φ thattracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliablecomputation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-basedmethods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.Read less <