This paper studies the ergodic rate performance of regularized zero-forcing (RZF) precoding in the downlink of a multi-user multiple-input single-output (MISO) system, where the channel between the base station (BS) and each user is modeled by the double scattering model. This non-Gaussian channel model is a function of both the antenna correlation and the structure of scattering in the propagation environment. This paper makes the preliminary contribution of deriving the minimum-mean-square-error (MMSE) channel estimate for this model. Then under the assumption that the users are divided into groups of common correlation matrices, this paper derives deterministic approximations of the signal-to-interference-plus-noise ratio (SINR) and the ergodic rate, which are almost surely tight in the limit that the number of BS antennas, the number of users, and the number of scatterers in each group grow infinitely large. The derived results are expressed in a closed-form for the special case of multi-keyhole channels. The simulation results confirm the close match provided by the asymptotic analysis for moderate system dimensions. We show that the maximum number of users that can be supported simultaneously, while realizing large-scale MIMO gains, is equal to the number of scatterers.