Based on our previously developed scheme to stabilize nonplanar optical resonant cavities utilizing polarization caused by a geometric phase in electromagnetic waves traveling along a twisted path, we report an application of the technique for a cavity installed in the Accelerator Test Facility, a 1.3-GeV electron beam accelerator at KEK, in which photons are generated by laser-Compton scattering. We successfully achieved a power enhancement of 1200 with 1.4% fluctuation, which means that the optical path length of the cavity has been controlled with a precision of 14 pm under an accelerator environment. In addition, polarization switching utilizing a geometric phase of the nonplanar cavity was demonstrated.