Strong gate control of proximity-induced spin-orbit coupling was recently predicted in bilayer graphene/transition metal dichalcogenide (BLG/TMDC) heterostructures, as charge carriers can easily be shifted between the two graphene layers, and only one of them is in close contact to the TMDC. The presence of spin-orbit coupling can be probed by weak antilocalization (WAL) in low-field magnetotransport measurements. When the spin-orbit splitting in such a heterostructure increases with the out-of-plane electric displacement field (D) over bar, one intuitively expects a concomitant increase of WAL visibility. Our experiments show that this is not the case. Instead, we observe a maximum of WAL visibility around (D) over bar = 0. This counterintuitive behavior originates in the intricate dependence of WAL in graphene on symmetric and antisymmetric spin lifetimes, caused by the valley-Zeeman and Rashba terms, respectively. Our observations are confirmed by calculating spin precession and spin lifetimes from an 8 x 8 model Hamiltonian of BLG/TMDC.