In juxtaposition to MP2 and its variants, Brillouin-Wigner theory employs an approach for generating corrections to the energy and wavefunction that differs structurally from RSPT. Its second order variant BW2 has seen little attention in the past century as it fails to be either size-consistent or size-extensive. Nonetheless, BW2 is a regular perturbation theory. Previous work focused on a modification for approximate extensivity , and a more rigorous quasiparticle QPMP2 approach to restore size-consistency and extensivity based on Green's functions. Our present work stems from another recent development that advantageously combines the advantages of BW2 and MP2 to create a size-consistent, sizeextensive and regular theory called BW-s2(α). This theory was able to achieve root mean square deviations (RMSD) of below 1.5 kcal/mol in various relative energy benchmark sets. Nonetheless, finetuning of BW-s2(α) is a rather limited process as a consequence of its singular parameter. In this work, we therefore seek to construct the most general expression of the regulariser in BW-s2(α) while maintaining its computational scaling as well as physical interpretability of the resulting zero-order Hamiltonian.