We explore in this paper the efficacy of the Rayleigh-Schrodinger (RS) and the Brillouin-Wigner (BW) perturbative counterparts of our recently developed multireference state-specific coupled-cluster formalism (SS-MRCC) with a complete active space (CAS). It is size-extensive and is designed to avoid intruders. The parent SS-MRCC method uses a sum-of-exponentials type of Ansatz for the wave operator. The redundancy inherent in such a choice is resolved by postulating suitable sufficiency conditions which at the same time ensure size-extensivity and size-consistency. The combining coefficients c(mu) for phi(mu)＇s are completely relaxed and are obtained by diagonalizing an effective operator in the model space, one root of which is the target eigenvalue of our interest. By invokation of a suitable partitioning of the Hamiltonian, very convenient perturbative versions of the formalism in both the RS and the BW forms are developed for the second-order energy. The unperturbed Hamiltonian is akin to the Epstein-Nesbet type when at least one of the orbitals is inactive and is the entire active portion of the Hamiltonian when all the orbitals involved are active. Illustrative numerical applications are presented for potential energy surfaces (PES) of a number of model and realistic systems where intruders exist and for molecules in their ground states with pronounced multireference character. Single reference MBPT and effective Hamiltonian-based multireference MBPT second-order results are also presented for comparisons. The results indicate the smooth performance of our state-specific perturbative formalisms in and around the region of intruders in the PES, indicating their suitability in bypassing intruders. In contrast, the effective Hamiltonian-based MBPT methods behave poorly in the regions of intruders.