Spin-torque nano-oscillators are promising candidates for many radio frequency and magnon-based nanodevices due to their broad frequency tunability, easy fabrication and high durability. To explore the tunability, we chose a [Ni/Co]/Pt-based spin Hall nano-oscillator with a moderate uniaxial anisotropy to systematically study the corresponding magnetodynamics excited by locally injecting a dc current into a nanoscale region of the extended multilayers [Ni/Co]/Pt under certain conditions. We find that the excitation current, the magnitude and orientation of magnetic field, and temperature can be used as a tool to selectively excite certain frequency bullet modes. The transition between nonlinear self-localized bullet modes with different frequencies is caused by the experimental parameter-induced change of energy landscape because, in the [Ni/Co]/Pt system, the strong spatial fluctuation of interfacial magnetic anisotropy leads to the variations of the internal magnetic field of the actual device. Our results demonstrate that the fluctuations of magnetic properties can promote experimental control of spin-torque driven magnetization dynamics in spin Hall nano-oscillators, and the application of expediting nonlinear magnetization oscillators in magnon-based devices and neuromorphic computing.