We present results of a theoretical study of influence of the electron velocity spread and the radial width on the efficiency and mode competition in a 300-kW, 300-GHz gyrotron operating in the TE22,2 mode. This gyrotron was developed for application to collective Thomson scattering diagnostics in the large helical device and 300-kW level high power single TE22,2 mode oscillation has been demonstrated [Yamaguchi et al., J. Instrum. 10, c10002 (2015)]. Effects of a finite voltage rise time corresponding to the real power supply of this gyrotron are also considered. Simulations tracking eight competing modes show that the electron velocity spread and the finite beam width influence not only the efficiency of the gyrotron operation but also the mode competition scenario during the startup phase. A combination of the finite rise time with the electron velocity spread or the finite beam width affects the mode competition scenario. The simulation calculation reproduces the experimental observation of high power single mode oscillation of the TE22,2 mode as the design mode. This gives a theoretical basis of the experimentally obtained high power oscillation with the design mode in a real gyrotron and moreover shows a high power oscillation regime of the design mode.