We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as Hamilton--Jacobi equations with random forcing. Understanding the large-time behavior of solutions is closely connected to fundamental questions concerning fluctuations and scaling in such growth processes. In this article, we establish, up to slowly varying factors, convergence rates with exponent $1/2$ for the large-time averages of both the solutions and the associated metric problem toward their ergodic limits. Our proof relies crucially on a new almost-Lipschitz regularity theory for the metric problem, which is of independent interest.