We develop an efficient approach for computing two-particle response functions and interaction vertices for multiorbital strongly correlated systems based on the rotationally invariant slave-boson framework. The method is applied to the degenerate three-orbital Hubbard-Kanamori model for investigating the origin of the s-wave orbital antisymmetric spin-triplet superconductivity in Hund's metal regime, previously found in the dynamical mean-field theory studies. By computing the pairing interaction considering the particleparticle and the particle-hole scattering channels, we identify the mechanism leading to the pairing instability around Hund's metal crossover arises from the particle-particle channel, which contains the local electron pair fluctuation between different particle-number sectors of the atomic Hilbert space. On the other hand, the particle-hole spin fluctuations induce the s-wave pairing instability before entering Hund's regime. Our approach paves the way for investigating the pairing mechanism in realistic correlated materials.