This article aims to derive novel varieties of exact solitonic wave solutions to the complex short pulse equation using an effective technique known as the Riccati-Bernoulli sub-ODE method (RBSODM), which does not conform to the balance rule. For the first time, the resonance induced by the arbitrariness of the singular manifold is analyzed in the proposed model through the application of Painlevé analysis (PA). The complex short pulse (CSP) equation models the behaviour of ultra-short optical pulses in nonlinear media. It serves as a more accurate model than the non-linear Schrödinger equation when the pulse width approaches the optical cycle scale. The proposed model incorporates both dispersion and Kerr-type nonlinearity, capturing the essential features of femtosecond pulse dynamics. Diverse types of rogue wave soliton solutions have been extracted such as bright soliton, dark soliton, W-like soliton, M-like soliton, and higher-order breather soliton. Moreover, the numerical approximations for all obtained analytical traveling wave solutions have been implemented by using the Haar wavelet approach (HWA). A comparison between the obtained analytical and numerical solutions is presented. Two-dimensional and three-dimensional graphical simulations are generated using the Mathematica software. The graphical simulations demonstrates the novelty of the obtained results and facilitate the interpretation of the dynamical properties of the proposed model.