We employ a high-quality linear axis-encircling electron beam generated by a Cuccia coupler to drive a Ka-band third-harmonic peniotron and develop a self-consistent nonlinear calculation code to numerically analyze the characteristics of the designed peniotron. It is demonstrated that through a Cuccia coupler, a 6 kV, 0.5 A pencil beam and an input microwave power of 16 kW at 10 GHz can generate a 37 kV, 0.5 A linear axis-encircling beam, and it is characterized by a very low velocity spread. Moreover, the electron beam guiding center deviation can be adjusted easily. Driven by such a beam, a 30 GHz, Ka-band third-harmonic peniotron is predicted to achieve a conversion efficiency of 51.0% and a microwave output power of 9.44 kW; the results are in good agreement with the Magic3D simulation. Using this code, we studied the factors influencing the peniotron performance, and it can provide some guidelines for the design of a Ka-band third-harmonic peniotron driven by a linear electron beam and can promote the application of high-harmonic peniotrons in practice.