Estimation of nonlinear dynamic models from data poses many challenges, including model instability and nonconvexity of long-term simulation fidelity. Recently Lagrangian relaxation has been proposed as a method to approximate simulation fidelity and guarantee stability via semidefinite programming (SDP); however, the resulting SDPs have large dimension, limiting their utility in practical problems. In this paper, we develop a path-following interior-point algorithm that takes advantage of special structure in the problem and reduces computational complexity from cubic to linear growth with the length of the dataset. The new algorithm enables empirical comparisons to established methods including nonlinear autoregressive models with exogenous inputs, and we demonstrate superior generalization to new data. We also explore the "regularizing" effect of stability constraints as an alternative to regressor subset selection.