We present the quantitative phase diagram of the bilayer bosonic fractional quantum Hall system on the torus geometry at total filling factor ν=1 in the lowest Landau level. We consider short-range interactions within and between the two layers, as well as the interlayer tunneling. In the fully polarized regime, we provide an updated detailed numerical analysis to establish the presence of the Moore-Read phase of both even and odd numbers of particles. In the actual bilayer situation, we find that both interlayer interactions and tunneling can provide the physical mechanism necessary for the low-energy physics to be driven by the fully polarized regime, thus leading to the emergence of the Moore-Read phase. Interlayer interactions favor a ferromagnetic phase when the system is SU(2) symmetric, while the interlayer tunneling acts as a Zeeman field polarizing the system. Besides the Moore-Read phase, the (220) Halperin state and the coupled Moore-Read state are also realized in this model. We study their stability against each other.