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Collinear equilibrium points organise important families of local trajectories in restricted three-body dynamics, but the inclusion of radiation pressure and oblateness complicates both their location and the construction of periodic approximations. This study examines the planar circular restricted three-body formulation adopted for the α-Centauri binary system, with dimensionless mass parameter μ = 0.47333, following the parameter convention of Singh and Gyegwe (2017), and radiating, oblate primaries. The equations of motion and collinear equilibrium conditions are stated in a rotating frame; local Taylor expansions about the three equilibria provide the basis for a Lindstedt–Poincaré construction that incorporates an amplitude-dependent time transformation. Particular attention is paid to the cubic-order Coriolis terms and the relation between time scaling and velocity reconstruction. Six reported combined parameter configurations are compared using equilibrium coordinates and benchmark starting conditions, periods, Jacobi constants and frequencies. For example, the reported L₁ equilibrium coordinate changes from −1.20751483 to −0.79423688 between the first two cases, while the corresponding values at the remaining equilibria also change non-monotonically. Because radiation and oblateness vary simultaneously, these differences cannot be attributed uniquely to either effect. Moreover, the numerical tables are benchmarks inherited from earlier work, not new integrations of the present approximation. The formulation provides a structured basis for examining perturbative consistency and designing subsequent numerical verification; confirmation of orbit closure, invariant preservation and predictive accuracy remains necessary.