This study presents a semi-analytical solution method to investigate the influence of an elliptical cutout in a composite cylindrical thin shell with non-uniform wall thickness and oval cross section on its buckling response under compression. This approach is based on the principle of minimum potential energy with local and global functions without the requirement of kinematic admissibility. The local functions capture the steep stress gradients and local deformations around the cutout. The kinematic boundary conditions on the upper and lower edges of the cylindrical shell are idealized by elastic edge restraints (supports). Specifying appropriate stiffness values for the elastic extensional and rotational springs allows the imposition of the kinematic boundary conditions. The non-uniform wall-thickness variation of the shell surface, which also leads to non-uniform material properties, is represented by perturbing the ply thicknesses by a periodical function either in the longitudinal or circumferential direction. The curvature of the non-uniform cross section is achieved by introducing an eccentricity parameter for an oval cross-section. The analysis involves two consecutive steps: (1) a pre-buckled state occurs, providing the membrane and bending stress distribution prior to buckling and (2) a buckled state caused by membrane stresses proportional to the reference (prebuckled) state. The present approach provides predictions in close agreement to those of high-fidelity finite element analysis.