Landau level (LL) gaps are important parameters for understanding electronic interactions and symmetry-broken processes in bilayer graphene (BLG). Here we present transport spectroscopy measurements of LL gaps in double-gated suspended BLG with high mobilities in the quantum Hall regime. By using bias as a spectroscopic tool, we measure the gap Δ for the quantum Hall (QH) state at filling factors ν = ±4 and -2. The single-particle Δ(ν=4) scales linearly with magnetic field B and is independent of the out-of-plane electric field E⊥. For the symmetry-broken ν = -2 state, the measured values of Δ(ν=-2) are ∼1.1 meV/T and 0.17 meV/T for singly gated geometry and dual-gated geometry at E⊥ = 0, respectively. The difference between the two values arises from the E⊥. dependence of Δ(ν=-2), suggesting that the ν = -2 state is layer polarized. Our studies provide the first measurements of the gaps of the broken symmetry QH states in BLG with well-controlled E⊥ and establish a robust method that can be implemented for studying similar states in other layered materials.