Analytical treatments of tunneling in bilayer graphene have typically relied on minimal models including only the vertical interlayer hopping γ₁ and have been restricted to the weak interlayer-bias regime (2ε ≪ γ₁). Consequently, they cannot adequately describe lattice deformations or strong electric-field effects. In this work, we present an analytical theory of evanescent states in electrically gapped bilayer graphene that overcomes both limitations. Our approach explicitly incorporates the skew interlayer hoppings γ₃ and γ₄ and remains valid even when the interlayer bias 2ε is comparable to γ₁. Focusing on low-energy electronic states near the charge neutrality point, we analytically derive the complex longitudinal wave numbers, the gap width, and the sublattice pseudospin within the electric-field-induced gap. We then systematically analyze the dependence of these quantities on the interlayer shear displacement δ⃗=(δₓ,δ_y), and find that skew interlayer hoppings, in particular γ₃, play an essential role. For transport along the zigzag (x) direction, the longitudinal wave vector becomes complex, whereas the transverse wave vector remains real. For a monolayer/bilayer/monolayer junction with transport along the zigzag direction, we find that δ_y has a significantly stronger impact on the conductance than δₓ. Furthermore, we identify a shear-induced phase proportional to δ_y that appears universally in the analytical expressions for the gap width, the sublattice pseudospin, and the decay length. These results establish a unified framework for shear- and bias-controlled evanescent tunneling in bilayer graphene and suggest broader relevance to nonequilibrium transport phenomena in layered materials.