We study spin transport in bilayer graphene (BLG), spin-orbit coupled to a tungsten di sulfide (WS2) substrate, and measure a record spin lifetime anisotropy similar to 40-70, i.e., the ratio between the out-of-plane tau(perpendicular to) and in-plane spin relaxation time tau(parallel to). We control the injection and detection of in-plane and out-of-plane spins via the shape anisotropy of the ferromagnetic electrodes. We estimate tau(perpendicular to) similar to 1-2 ns via Hanle measurements at high perpendicular magnetic fields and via a new tool we develop: oblique spin valve measurements. Using Hanle spin-precession experiments we find a low tau(parallel to) similar to 30 ps in the electron-doped regime which only weakly depends on the carrier density in the BLG and conductivity of the underlying WS2, indicating proximity-induced spin-orbit coupling (SOC) in the BLG. Such high tau(perpendicular to )and spin lifetime anisotropy are clear signatures of strong spin-valley coupling for out-of-plane spins in BLG/WS2 systems in the presence of SOC and unlock the potential of BLG/transition metal dichalcogenide heterostructures for developing future spintronic applications.