We report a detailed study of the microscopic effects of Cu intercalation on the charge density wave (CDW) in 1T-CuxTiSe2. Scanning tunneling microscopy and spectroscopy reveal a unique, Cu-driven spatial texturing of the charge-ordered phase, with the appearance of energy-dependent CDW patches and sharp π-phase shift domain walls (πDWs). The energy and doping dependencies of the patchwork are directly linked to the inhomogeneous potential landscape due to the Cu intercalants. They imply a CDW gap with unusual features, including a large amplitude, the opening below the Fermi level, and a shift to higher binding energy with electron doping. Unlike the patchwork, the πDWs occur independently of the intercalated Cu distribution. They remain atomically sharp throughout the investigated phase diagram and occur in both superconducting and nonsuperconducting specimens. These results provide unique atomic-scale insight into the CDW ground state, questioning the existence of incommensurate CDW domain walls and contributing to understanding its formation mechanism and interplay with superconductivity.