Harmonic multiplying gyrotron traveling-wave amplifiers (gyro-TWAs) permit for magnetic field reduction and frequency multiplication. A high-order-mode harmonic multiplying gyro-TWA with large circuit dimensions and low ohmic loss can achieve a high average power. By amplifying a fundamental harmonic TE01 drive wave, the second harmonic component of the beam current initiates a TE02 wave to be amplified. Wall losses can suppress some competing modes because they act as an effective sink of the energy of the modes. However, such wall losses do not suppress all competing modes as the fields are contracted in the copper section in the gyro-TWA. An improved mode-selective circuit, using circular waveguides with the specified radii, can provide the rejection points within the frequency range to suppress the competing modes. The simulated results reveal that the mode-selective circuit can provide an attenuation of more than 10 dB to suppress the competing modes (TE21, TE51, TE22, and TE03). A G-band second harmonic multiplying gyro-TWA with the mode-selective circuit is predicted to yield a peak output power of 50 kW at 198.8 GHz, corresponding to a saturated gain of 55 dB at an interaction efficiency of 10%. The full width at half maximum bandwidth is 5 GHz.