In this work, we performed a detailed scanning tunnelling microscopy and spectroscopy study of the charge density wave state of 1T-CuxTiSe2 in a wide range of Cu concentrations (0 < x < 0.07), an apparently perfect playground for investigating the interplay between correlated ground states. Thanks to density functional theory modelling, harmonic wave simulations and defect analysis, we were able to resolve for the first time the charge density wave contributions from the full Se-Ti-Se trilayer with subatomic precision, and to show the excellent agreement with the proposed periodic lattice distortion models. By changing the energy integration windows in our measurements, we reported perfect contrast inversion of the charge density wave signal, as expected by a simple mean-field BCS-like theory and by density functional theory modelling. Conversely, we found a mismatch in the energies at which this contrast inversion takes place between the theoretical value (Eci=EF) and the experimental one (Eci ~ -200 mV), which together with spectroscopic data suggest that in this system the charge density wave formation does not yield a gap opening at the Fermi energy, but at slightly higher binding energies. Moreover, the gap position is extremely sensitive to external perturbations such as surface adsorbates and chemical doping. In fact, by adding Cu atoms to the system, we reported an unprecedented energy-dependent patchwork of charge density wave regions, closely related to the local Cu content, to the consequent changes in the potential lanscape and, accordingly, to the charge density wave gap position. The reported resilience of the charge density wave gap further allows the quantum condensate to endure strong external perturbations, such as chemical doping, the loss of long-range order induced by exciton melting in metallic samples and the formation of superconductivity.