Bilayer graphene (BLG) was recently shown to host a band-inverted phase with unconventional topology emerging from the Ising-type spin-orbit interaction (SOI) induced by the proximity of transition metal dichalcogenides with large intrinsic SOI. Here, we report the stabilization of this band-inverted phase in BLG symmetrically encapsulated in tungsten diselenide (WSe2) via hydrostatic pressure. Our observations from low temperature transport measurements are consistent with a single particle model with induced Ising SOI of opposite sign on the two graphene layers. To confirm the strengthening of the inverted phase, we present thermal activation measurements and show that the SOI-induced band gap increases by more than 100% due to the applied pressure. Finally, the investigation of Landau level spectra reveals the dependence of the level-crossings on the applied magnetic field, which further confirms the enhancement of SOI with pressure.