A major difficulty in aerodynamic design is related to the multiplicity of geometrical representations handled during the optimization process. From high-order Computer-Aided Design (CAD) objects to discrete mesh-based descriptions, several geometrical transformations have to be performed, that considerably impact the accuracy, the robustness and the complexity of the design loop. This is even more critical when Multiphysics applications are targeted, including moving bodies. To overcome this limitation, we present here a fully integrated methodology that combines ideas from Iso-Geometric Analysis (IGA) and high-order Discontinuous Galerkin methods, within an Arbitrary Lagrangian-Eulerian (ALE) formulation for compressible Navier-Stokes equations. The resulting approach relies on a single geometrical representation, based on Non-Uniform Rational B- Splines (NURBS), that allows to achieve geometrically exact simulations with adaptive local refinements and moving interfaces. We consider as demonstration the optimization of a morphing airfoil. The profile is defined by two cubic NURBS curves, whose deformations are controlled in terms of frequency, amplitude and length, while a C1 regularity is maintained during the whole process. A multi-objective Bayesian optimization is carried out to select the most promising control parameters with respect to the time-averaged lift, the instantaneous minimum lift and the time-averaged actuation power.