Recent infrared measurements of phonon peaks in gated bilayer graphene reveal two striking signatures of electron-phonon interaction: an asymmetric Fano lineshape and a giant variation in the peak intensity as a function of the applied gate voltage. In this Rapid Communication we provide a unified theoretical framework which accounts for both these effects and unveils the occurrence of a switching mechanism between symmetric (Eg) and antisymmetric (Eu) phonon modes as dominant channel in the optical response. A complete phase diagram of the optical phonon response is also presented, as a function of both the charge density and the band gap.