Energy Spectrum and Gauge Transformations

The time-independent Schrodinger equation $\hat{H} \psi = E \psi$ only holds when the Hamiltonian does not depend explicitly on time. If you start with a time-independent Hamiltonian and make a time-dependent gauge transformation, then the new Hamiltonian will depend explicitly on time, and there is no reason to expect that the (time-dependent) eigenvalues of $\tilde{H}$ will correspond to any kind of physical energy.