Incommensurate relaxed overlayer Moiré structures are often interpreted as systems with weak lateral variations of the binding potential and thus no structural modulations in the overlayer material. We discuss here the example of a CoO(111) bilayer on Ir(100), which is a relaxed overlayer with strong structural response to the lateral modulation of interface properties but nevertheless is incommensurate. By means of DFT calculations we quantitatively reproduce all the structural parameters of the CoO(111) bilayer on Ir(100) as proposed by a recent LEED analysis [C. Ebensperger et al., Phys. Rev. B 81, 235405 (2010)]. The calculations predict energetic degeneracies with respect to registry shifts of the CoO(111) film along [011]. Large-scale, low-temperature STM topographies reveal that the true structure of the film is incommensurate in this direction, exhibiting a onedimensional Moiré pattern with a period of about 9.4 aIr. From DFT calculations for limiting (periodic) models, we can sample the potential landscape of the cobalt and oxygen atoms in the Moiré structure across the Ir(100) unit cell. We find that despite the non-commensurability of the film, the binding to the substrate is site specific with strong attraction and repulsion points for both cobalt and oxygen atoms, leading to severe local distortions in the film. The lateral modulation of the structural elements within the oxide film can be understood as a combination of the lateral variation in the Co-Ir binding potential and additional O-Ir binding.