A magnetic cusp gun (MCG) is being developed to generate an axis-encircling electron beam, which is called the large orbit beam, which is going to drive a 0.396-THz fourth-harmonic gyrotron. Developing an MCG imposes crucial challenges on a simultaneously minimizing guiding center deviation and velocity spread of the electron beam, particularly because an ultrahigh magnetic compression ratio is unavoidable, as is the case for a terahertz (THz) gyrotron. The study of the electron dynamics in the MCG reveals that, close to the emitter, a pair of focusing electrodes are employed to construct a special focusing and accelerating electric field as a way to balance the space-charge influence and guiding center deviation. Investigation indicates that both the electron-beam generalized-angular-momentum spread and the guiding center distribution are the critical factors contributing to beam parameter spread. Intensive optimization generates a high-power MCG with a pitch factor of 1.5, the highest magnetic field of 4 T, minimum transverse velocity spread of 1.1%, and a beam current of 2 A. The key parameters exhibit excellent stability tuning over a wide range of beam current and magnetic field. These merits enable the harmonic gyrotrons or even the frequency-tunable THz gyrotrons to be developed.