Controlling magnon currents is central to the development of magnonic devices, which exploit spin excitations for information transport without charge flow. Here, we use the spin-orbit torque (SOT) from a Pt gate to control the effective magnon damping in three-terminal devices fabricated from low-damping lithium aluminum ferrite (LAFO) thin films and identify the nonlinear damping that ultimately limits the gate response. With increasing gate current, the nonlocal magnon signal initially increases, but saturates and subsequently decreases above a critical gate current density. Brillouin light scattering microscopy reveals that this turnover coincides with the emergence of an SOT-driven auto-oscillation mode localized beneath the gate, whose frequency continuously redshifts with increasing gate current, characteristic of large-amplitude nonlinear dynamics, and with a suppression of the propagating magnon population away from the gate. Micromagnetic simulations reproduce the observed nonmonotonic gate response only when an amplitude-dependent effective damping is included, indicating that nonlinear losses, rather than saturation of spin injection, limit the degree of magnon modulation. Our results show that SOT compensation of magnon damping is ultimately limited by nonlinear damping attributed to auto-oscillations, providing a framework for understanding the limits of magnon current modulation in magnetic insulators.