The two-mode quantum Rabi model with bilinear coupling is studied using extended squeezed states. We derive $G$-functions for each Bargmann index $q$% . They share a common structure with the $G$-function of the one-photon and two-photon quantum Rabi models. The regular spectrum is given by zeros of the $G$-function while the conditions for the presence of doubly degenerate (exceptional) eigenvalues are obtained in closed form through the lifting property. The simple singularity structure of the $G$-function allows to draw conclusions about the distribution of eigenvalues along the real axis and to understand the spectral collapse phenomenon when the coupling reaches a critical value.
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