The transmission and reflection of finite amplitude, three-dimensional, internal gravity wave packets across a reflection level are examined numerically in a Boussinesq fluid with a nonuni- form retrograde shear flow. We derive the critical amplitude for wave packets to transmit partially above the reflection level predicted by linear theory, occurring when the magnitude of the vertical shear associated with their wave-induced mean flow is locally greater than that of the background shear. We propose a novel weakly nonlinear mechanism, based on self-induced local decreases in buoyancy, to explain the generation of secondary wave packets by nonbreaking primary waves, and predict the critical amplitude for its onset. Numerical simulations are performed for a range of nonhydrostatic wave packets with small to mod- erately large initial amplitudes with their predicted wave-induced mean flow superimposed. Transmission is quantified using the pseudomomentum corresponding to upward-propagating waves above the reflection level predicted by linear theory. In most cases the transmission transiently grows and decays as wave packets first cross and then reflect from the reflection level predicted by linear theory. We find that for all but the most strongly nonhydrostatic wave packets, larger-amplitude waves exhibit smaller peak transmission, relative to the to- tal pseudomomentum. The time at which peak transmission occurs is diagnosed. Strongly nonhydrostatic wave packets exhibit continuous transmission well above the reflection level. When we consider the time interval for transmission to decrease to half its peak value, we find this becomes longer with larger initial amplitude. These behaviours are found to result from the combined effects of modulational instability and the generation and evolution of secondary wave packets. Results are discussed in the context of previous studies of one- and two-dimensional wave packet transmission and reflection.