We investigate the electronic band structure and the proximity exchange effect in bilayer graphene (BLG) on a family of ferromagnetic multilayers Cr2X2Te6, X = Ge, Si, and Sn, with first principles methods. In each case the intrinsic electric field of the heterostructure induces an orbital gap on the order of 10 meV in the graphene bilayer. The proximity exchange is strongly band-dependent. For example, in the case of Cr2Ge2Te6, the low energy valence band of BLG has exchange splitting of 8 meV, while the low energy conduction band's splitting is 30 times less (0.3 meV). This striking discrepancy stems from the layer-dependent hybridization with the ferromagnetic substrate. Remarkably, applying a vertical electric field of a few V nm(-1) reverses the exchange, allowing us to effectively turn ON and OFF proximity magnetism in BLG. Such a field-effect should be generic for van der Waals bilayers on ferromagnetic insulators, opening new possibilities for spin-based devices.